The Mayo Clinic has an article on the topic from Dr. Edward R. Laskowski, M.D. that goes over the possible cause:
Hand swelling during exercise is a fairly common problem. The cause
isn't completely clear, but hand swelling appears to be a result of
the way your body and blood vessels respond to the increased energy
demands of your muscles during exercise.
A possible remedy is proposed by this article on Real Simple, though aimed at joggers, it may apply to you - do the hand exercises between sets:
“Raising your arms over-head and pumping your fists can help keep
blood flowing as you jog,” says Sosena Kebede, an assistant professor
of medicine at Johns Hopkins University School of Medicine, in
However, both articles indicate the swelling should go away a few hours, not days, after you've stopped exercising. This may be something you'd want to discuss with a healthcare professional.