I would have to know a lot more information and be able to see your tongue to give anything more than I can offer here, but I will offer what I can.
First self medication is always a bad idea with Chinese herbs. Most of this is because of the complexity of the diagnostic system that helps you choose the correct formula, but the other main part is how often the formulas will change as you change (ideally as you improve). Chinese herbal medicine needs constant guidance and tweaking to be its most effective. All that said, si ni tang wan would very likely be low on my list of formulas for someone with hashimoto's. The reason for this is you have to watch stronger tonifying herbs with autoimmune conditions like hashimoto's.
The general issue with hashimoto's is that it is quite complex. It will generally require acupuncture and herbal medicine in many cases for 6 months to 2 years (usually not weekly acupuncture, but very consistent treatment). If you cannot afford your current practitioner, I strongly suggest you find one that you can and one that doesn't upsell on the herbal medicine side. If you are relatively happy with your current practitioner but just can't afford the herbs, ask him what specific formulas he was going to use and if he has a problem with you finding them elsewhere. Herbal medicine is extremely affordable so long as people are not strongly marking up the herbs.
Along with acupuncture and herbal medicine, tong ren therapy (the energy healing system we use) can be quite helpful and is generally obtainable at a distance for free. A response to someone with hashimotos describing our general protocol is here. You can obtain treatment for free via many online classes at tongrenstation.com or via our weekly broadcast currently on tuesdays at 6pm. Incidentally the same points we use for tong ren are used during the acupuncture treatments as well in most cases along with others that are more directly tied to the persons individual diagnosis.