Hi Beth819,
That’s great news 
It’s always a good idea to use a mattress protector on any mattress to protect against stains and accidental spills (which can void a warranty) and to protect the mattress from the normal body fluids and oils that we release each night and from skin particles that can contribute to dust mite populations (they feed on skin particles). Having a protector that can easily be removed and washed will help keep your sleeping surface in a clean and hygienic condition. There are several different types of protectors and each of them have their own pros and cons so which protector is best would depend on the criteria that are most important and the preferences of each person but they basically boil down to three types (four if you consider the vinyl types which don’t breathe at all and I wouldn’t consider).
The first are the thinner ones that have a membrane in them which are waterproof and also somewhat breathable. I’m guessing this is the type you have. Some of them have more of a “shower curtain” feeling to them than others but the better ones seem to work fairly well and most people don’t find them to feel like vinyl or plastic although there are some who don’t like them. If they are too tight they can create a drum effect on the mattress surface. The better ones have the membrane attached to a cotton terry or better yet a stretch knit fabric and are fairly stretchy. Which brand do you have?
Two of the most popular of these are the Luna and the Protect-A-Bed. Gotcha Covered has an organic stretch knit version here and Naturepedic also has an organic flannel version here. The Protect-A-Bed luxury protector uses a stretch knit Tencel fabric (Eucalyptus) and is Oeko-Tex certified. A few more similar protectors in this category are listed at the end of post #15 here and in post #1 here. The advantage of the better ones of this type are that they are fairly thin and stretchy so they don’t affect the feel of the mattress much and they are also waterproof. Their disadvantage is that the membrane is not as breathable as other types of protectors and can contribute to sleeping warmer for some people. Some of them also have the “crinkly” feel as you mentioned that some people may notice more than others. I would also make sure that you follow the care instructions carefully with these so that they don’t melt or lose their waterproof properties when you clean them. Some of these also have waterproof protection on 5 sides rather than just the top surface (see post #2 here) which can help protect the mattress from spills or accidents that run down the side of the mattress although using one of these would have the tradeoff that the mattress can be less breathable and allow less airflow than using a protector that is only waterproof on the top surface.
The next type is a thinner wool mattress protector. Some of these have a cotton cover with a thin layer of wool inside. Others are wool puddle pads or moisture pads with felted or tightly woven or needle punched wool to make the wool water resistant but these do best with an absorbent sheet or a fitted cotton protector above them to absorb and spread out some of the moisture and add additional protection. Wool is very breathable and can help to cool down a mattress and control humidity and temperature. The cotton wicks moisture very well into the wool. These are only water resistant however (not waterproof) which for most people is fine. They are also a little thicker so they will have a little more affect on the mattress comfort layers.
One of the most popular of the cotton/wool protectors is the St Dormier although many of the online descriptions for this are incorrect and it has 4.5 oz/yd2 rather than the 7.5 oz/yd2 which many of the online descriptions indicate which may be why they have less effect on the “feel” of a mattress. Natura also makes one that has both a thinner layer of wool and a breathable membrane called the NaturaProtect here for those who want wool and a breathable/waterproof membrane but this would not be quite as breathable as wool alone. They also have a washable wool mattress pad that has 10 oz/sq yd of washable wool and an organic version (the picture here may be wrong) with 6 oz/sq yd of organic wool (not washable) without the membrane. Glideaway also has a cotton/wool mattress protector with a waterproof membrane here and Sleep & Beyond also has a cotton/wool protector with a waterproof membrane here which appears to have about 5 oz/yd of wool. Suite Sleep has a cotton/wool mattress protector and 45th Street Bedding also has one here that uses 9 oz/yd of wool without a membrane as well. Foam Sweet Foam also has one that is 1/2" thick with no membrane here. Thicker layers of wool in the protector may have more of an effect on the “feel” of a mattress but will also have more of the temperature regulating properties of wool.
Some examples of felted or tightly woven needle punched wool “puddle pads” that can be used directly under a sheet or in combination with a stretch cotton protector (for some additional protection) are here and here and here and here and here and here and here and there are two DIY versions here and here and here. These can be more water resistant (especially with a thicker cotton sheet or mattress protector) and are quite popular for those who are more focused on either natural or organic materials (depending on the wool that is used) but they not completely “waterproof” and they are also more costly and can affect the “feel” of the mattress more than a thinner protector.
The third main type is a cotton fabric protector. These are like a thick sheet that can absorb moisture and body oils but they are not water resistant like wool. If they are made as a stretch knit (like a jersey sheet) … they can be very flexible and breathable and comfortable and will have little affect on the feel of the mattress. Some high quality examples of this include the CozyPure protector here and the Berkeley Ergonomics protector here, and the Suite Sleep protector here, and the Naturepedic cotton flannel protector here and the Sleeptek protector here (also sold under the Sueno and Naturally Organic brand names at different stores) and the Purerest protector here and the Lifekind/OMI protector here and the Green Sleep Vasilo mattress protector here.
All of these are thinner protectors and are designed to have less effect on the mattress than thicker mattress pads that include various types of padding as part of their design. I would also make sure that you do some google searching for each one since there can be a wide range or prices at different stores or online retailers.
So the choices between mattress protectors are the membrane type … the wool type … or the cotton type. The tradeoffs involved are between how waterproof they are … how breathable they are … how much they will affect the feel of your mattress, the importance of natural materials, and of course cost. There is more about the choices and tradeoffs involved and the amount of “protection” that may be important in post #2 here and in post #5 here.
What you have is a mattress pad which is thicker than all of these protectors and is more like an extra padding layer on the mattress itself which can be used to add some surface softness or fine tune the feel of a mattress. These will also have much more effect on the feel and performance of the mattress and can reduce how much you sink in to the latex or other foam layers below them and on how well they can contour to the shape of your body. The thicker and less stretchy the mattress protector or pad is … the more it can affect the latex or other softer foam materials underneath it. Thicker layers above memory foam can also reduce the amount of body heat that reaches the memory foam and reduce its ability to soften with body temperature. Some mattress pads are also water resistant or waterproof and can also double up as a mattress protector as well. There is more about different types of mattress pads in post #10 here.
Finally you have toppers which are even thicker than mattress pads and are usually only used to add softness to the mattress and not protection (you would usually still need a protector with a topper).
You would probably do best with some kind of protector (not a mattress pad) and which is best depends on which combination of water resistance, breathability, and effect on the mattress that would be best for you. If you get the membrane type … then some of them have cotton on both sides of the membrane which may be enough to take away the feel of the cover… Otherwise a thicker wool or cotton protector may be better.
For those that have significant allergies then a mattress encasement that completely surrounds the mattress may be worth considering (even though some of them may reduce the ability of the mattress to breathe and ventilate) to protect against dust mite allergens (and/or bed bugs if this is an issue). There is more about mattress encasements in post #2 here and about dust mite allergies in post #2 here.
Phoenix