Hi wildsea,
Welcome to the Mattress Forum! 
I’m sorry to hear about your Hashimoto’s. I was reading up on it and unfortunately there seems to be a relationship between Hashimoto’s Disease (and other autoimmune diseases) and increased sensitivities/allergies. I know that must be frustrating for you.
While a true latex allergy would be quite rare, it certainly seems that you’re reacting to your topper. While it is a complex subject, the essence of it is that there are basically 3 types of latex allergy or sensitivity (although two of them aren’t really an allergy to latex).
The first of these is irritant contact dermatitis which usually comes from an actual irritation to the skin such as from repeated donning of latex gloves or exposure to the powder on them often in combination with skin sweating or frequent hand washing. This is a result of physical skin irritation and wouldn’t apply to latex foam.
The second and most common of the “latex allergies” contact dermatitis allergies (type IV latex allergy) which is most likely a reaction to the chemicals additives or substances that are added to the latex in the manufacturing process and not the latex itself. This is more prevalent for those who have handled latex in various occupations over a period of time (such as health care workers using latex gloves). This is the less severe of the types of “latex allergies” and while it is not “common” it is also not so rare … particularly among certain occupations. Because latex foam has had most of the surface proteins and any remaining unreacted materials washed away during manufacturing (unlike other types of non foamed latex like gloves, condoms, balloons etc.), this is far less common with latex foam. This type of “latex allergy” is a contact allergy (or sensitivity) and the fact that most of the surface proteins and additives are washed away and that the latex is covered makes this type of allergy much less of an issue with latex mattresses. There is more detailed information and links in this thread and latex foam is not mentioned in any of the literature I’ve seen or linked as a possible source of this type of allergy.
The last type of latex allergy is type 1 which is much more rare but much more serious. This type of allergy can cause an anaphylactic reaction with breathing difficulties and swelling and like a bee sting in a small portion of the population can be life threatening. These people are almost always very aware of their allergy and have to go to great extremes to avoid natural latex which is very difficult because it is so widely used in our society. They for example have to check in restaurants where they eat that the workers aren’t wearing gloves with latex or they could have a severe reaction. Most people that sell latex mattresses can go a lifetime without ever seeing this type of latex allergy. This would be connected to the proteins found in natural latex which would be missing in synthetic latex.
There is more information and a brief description of each of these here. This is a very useful page with quite a bit of information. There is also much more information about latex allergies in this topic and the many other sources of information that it links to as well.
My first step would be a trip to your health care professional and describe to them your symptoms and have them perform an allergy test on you, as you’d want to be specific about any items that could potentially be causing you these issues. Assuming that latex is an issue for you, there are other topper options, and many good sources of which I am aware are listed in post #4 here. You may wish to look into a microcoil or buckling column gel topper, as those would tend to have less chance of causing you to have a reaction. There is also a link there for wool toppers, which are not as soft as latex but they can provide extra plushness. Those would all be good alternatives.
I’ll look forward to learning of your progress and hopefully you’re able to find something that doesn’t cause you to have a reaction, but still offers you some comfort.
Phoenix