Goodness! Sometimes you just have to tell your doctor what you want and don't take no for an answer. Go online and find the website of the hospital you use and see if you can compare the profiles of the doctors within the endocrine department or call that department and ask the secretary who she may recommend (if you are lucky, they will help). If you have no insurance, call the hospital business office to work out a payment plan and/or go to the social services of the hospital to see if you can qualify for help paying for meds.
If YOU know what works for you, just tell him/her!
Please post whatever thyroid test results and reference ranges you have, so that members can give you the best response. I have a suspicion that your doctors have been diagnosing and medicating you based on TSH, which frequently doesn't work very well at all.