Good morning!
First, a few questions:
If your pain is on the thumb side of your hand/palm, it is possible that it is CTS. Do your fingers tingle or go numb? Pain at night? I would suggest seeing your Dr. to get a proper dx.
~ Does your job or perhaps a hobby have you doing repetitive movements? (like gardening or a lot of computer work)
~Are your hamstrings tight?
~Where are the creases of your elbows in Down Dog? (should face each other)
~Has your teacher looked at your down dog and provided feedback on your alignment?
~Where are the creases of your wrists in down dog? (should be parallel to mat)
Depending on your answers, here are a few suggestions:
~Have your teacher assess you in down Dog.
~Wear a wrist brace. These can be purchased in a drugstore.
~Make sure your fingers are spread wide. Middle finger should point straight ahead. Press firmly into the pads of the fingers. A bit more so with thumbs and index fingers. Equal weight on all four corners of palm.
~If you lack hamstrings flexibility, your weight is likely shifting to your wrists to compensate. More than likely it’s the heels of the hands. Work on lengthening hamstrings.
~You can also try a slant wedge under your hands with the slanted/shorter side away from you.
Most importantly, if it is CTS, you need to rest the inflamed tissue. Then proceed with caution as you add back weight bearing poses. When the upper body/shoulder area is weak, we tend to let our weight fall more into the heels of the hands. Dolphin is a good modification until your strength is built back up and the inflamed tissue is healed.
When you are ready, here is a sequence I teach:
Hands in namaste. Drop hands down, then back up. Point fingers down, then back up. Keeping elbows/forearms straight, shift hands (still in namaste) to R. Then to L. Stay in each of these positions for several breaths.
Hope some of this helps.