Could be carpal tunnel syndrome but a few comments to make:
1. Never heard of it being a communicable disease. I think the fact that your mother had it too is a furphy. It is usually a primary problem, ie no cause, but can be secondary to other conditions such as thyroid disease, pregnancy, hormone abnormalities etc
2. Can't think of a connection between CTS and swelling under the armpits.
Basically, carpal tunnel syndrome is caused by compression of the median nerve as it enters the hand. This nerve supplies the skin of the thumb, index, middle and half of the ring finger, so the tingling in these fingers could be related. With chronic CTS, you also get wasting of the thenar eminance (this is the ball of muscles at the base of the thumb). Compare sides and see if one is more wasted than the other.
One question - do you wake at night with tingling in the fingers? If you do, wearing wrist splints might help
To be certain, nerve conduction studies can be done if there is any doubt about the diagnosis.
Don't think swimming will be a problem for you. If you have ongoing concerns you should go back to your local doctor.