I had it with both pregnancies (from 20 weeks) and in both arms. Here is what to do:
1) Buy wrist braces now. The good kind with metal inside them (about $20 each). I bought mine at CVS. Wear them ALL day and night, except when showering of course. The longer you wear them, the better it will be. Even if it doesn't hurt in the day, WEAR THEM.
2) At night before going to bed, put cold packs on your wrists and lower arms to reduce swelling. Get your husband to massage them. If you have any knots in your back, then putting a heating pad on them at this same time. It is all connected together so reduce the tightness in your back and shoulders and reduce the tightness and swelling in your arms and wrists. I would do this while watching TV in bed the hour before going to sleep.
3) Now this is the luxury part - Find a massage therapist who knows about treating it. I have one that lives in Dallas if you want the number. It helped a lot to have the back and arms massaged and she knew just the right spots.
I hate to tell you that it might last a little while after the baby is born, especially in the hospital when you body is getting rid of all those fluids. Be prepared to have lots of pillows to support the baby when you are feeding so you don't have to use your hands and arms to hold as much. Mine lasted weeks after, but I think that is because it was so bad during the pregnancy.
Good luck! If you need more advice, let me know. And count yourself lucky that you are already 37 weeks! :)