Take your current bike to the same LBS and ask them to put your bike on the trainer there. Ride for a few minutes while the staff looks at your riding position. That will tell you more about how to correct the issues.

Also, +1 on seeing an orthopedist who focuses on sports injuries and hands, specifically. Every region/city has several of these. Figure out what the medical/structural cause of your injury is, and then trace that back to the bike riding position that you currently have.

My guess is that the bike type and fit are both incorrect for you. But that's a guess...

You have many options to correct your problem. If it's just a fit issue, then you might be able to make your current bike work. Might be.