Hi Brigabart,
The better options I’m aware of in the Raleigh, NC area are listed in post #6 here.
I would call the ones that seem to make or carry mattresses that interest you and have a more natural or organic focus and talk to them about your needs on the phone and then visit the ones that you are most interested in trying or talking to.
There are also online options available but you won’t be able to test these in person so while they may provide a good value reference I would focus first on local options.
This is a very incomplete list with some examples of some manufacturers that make mattresses similar to what you are looking for (innerspring/natural fibers with no foam) to give you some ideas and a sense of the wide variety of designs and pricing available. Some of these include latex which you may want to exclude but you can look on their sites to see what they have available.
mygreenmattress.com (a member of this site)
https://www.gardnermattress.com/ (a member of this site)
http://www.vivetique.com/ see here as well www.daxstores.com/
Lifekind® Certified Organic and Naturally Safer® Since 1997 and their sister company http://www.omimattress.com/
http://www.royal-pedic.com/default.asp
http://sleeptek.ca/ see here Organic Mattresses and Bedding made by Sleeptek for an example
http://www.landandsky.com/index.asp see here for examples http://www.thenaturalbedstore.com/organic-mattress.asp
http://www.donsmattress.com/home.html Medford, MA. Small company making cotton mattresses.
https://mcroskey.com/ San Francisco, CA
There are also some “super premium” mattresses that use innersprings and natural fibers in their construction … some of which can go well into mid 5 figures. these include manufacturers such as VI Springs … Hastens … Savoir … Relyon … and Hypnos.
There are also some “old style” manufacturers in various areas of the country that still make innerspring/natural fiber mattresses as well but “foam free” mattresses certainly aren’t nearly as common as mattresses that contain some type of foam.
Hope this helps … and you should at least have some interesting reading to do 
Phoenix
PS: I should have mentioned as well that in many cases innerspring/fiber mattresses are on the firm side, particularly as the fibers compress, and if you want to add a wool topper for some extra softness and to relieve pressure points then post #3 here has some of the better sources for these I’m aware of.