SAMe does not have aspirin or NSAIDs in it.
Take the thyroid, wait 1/2 hour and then take the lipoic and SAMe. Since you are fluctuating with the levothyroxine, take it alone, for insurance. With a full glass of water it should leave the stomach in about 1/2 hour and move along.
The SAMe does not dissolve until the intestines. It has a special coating. That takes time, so the thyroid should have moved along by then. You can take the lipoic, B12 and SAMe together.
Wait another 1/2 hr or so before eating.
SAMe is made in the body from homocysteine, using the 3 vitamin cofactors and glycine
(which supplies the methyl group).
Once it gives up the methyl group, it is regenerated using the 3 B vitamins over and over. Some people fail in this step with age, so taking it as a supplement repairs that.
I'd start with a small package of 200mg... a 30 pack. Take one a day for a month. See how you feel. You should notice mood effects within the first week. But the effects on joints may take a month or two and then are slowly progressive in a positive way.
This link explains the chemistry:
It gets complicated. But... if you do respond well to SAMe, it might point to a DNA test for the MTHFR problem for you.
This link also shows the interlinking of the methylation chemistry to each other.