I have thumb pain just like you have. It was manageable when I'm riding my road bike because not too much pounding. It used to get real bad when I rode my mountain bike alot. Opening cans with a can opener was horrible. I went in the hand ortho doc and he took xrays and says I have thumb joint (dont know the technical term) instability. The bones overlap and rub. This was really bad for a couple of years. I avoid some things. I bought an electric can opener etc. For some reason, last year this started to get better, i.e. it does not hurt as much. I'm not sure why? I rode my road bike a lot. I don't like the shifters where you have to push the lever on mountain bikes, that will hurt my hand.

Oh, the doc's recommendation was to have surgery to take a ligament/tendon?? out of another place in my body and place it there at the thumb joint. Takes forever for it to heal and I just opted out of that since there was no way I wanted to be not able to use my hand for like at least a 6 month recuperation period!

I hope this is not what you have, but it could be. You can try different shifting on your bikes, i.e. grip shift, be careful about how you position your hands when you ride, etc.