You're not alone mama! I started getting the same pain right after my last baby was born, 4 months ago. My doc diagnosed DeQuervain's tendonitis. He sent me to a therapist who said that it's VERY common in pregnant/post partum women. I guess the hormones released to loosen up the muscles/tendons in the pelvic cavity for birth affect all muscles/tendons, and since mama's use there hands a lot,,, Anyway, the brace thing that Carol recommended helps A LOT, as does alternating cool/hot soaks. (Soak in cool water for 2 min, then hot for 2 min, alternating for about 10 min). Ice is wonderful,,, mine felt 90% better for about an hour after each icing, but be careful not to overdue it. After trying therapy, the brace, exercises, etc for 3 months my doc finally agreed to a local injection of cortisone with lidocain. He warned it would probably be a temporary fix, and that the next step would be a orthopod. But I lucked out and haven't had a problem since! You might ask your doc about the injection since it's localized and not injected into the blood stream.
GOOD LUCK!!!
P.S. carpal tunnel in the wrist generally effects the index and middle finger, not the thumb.