I've got a mild pain and swelling in my right hand's palm that I believe is a sprain. How to avoid aggravating my injury while continuing the training?
Don't continue the training. Stop.
Think about it: your body is injured and needs time to heal. If you don't give it time, it won't heal.
It is MUCH better to not train for a week than to have unhealed damage which, after a few years, becomes much worse.
First - the obvious stuff for sprains:
The second part gets a little tricky, because it means you have to adapt a lot of what you're training. It will, however, improve the speed of recovery and the final results.
Second, the more involved aspect for sprains:
After 3-4 days of initial healing, your body begins growing new tissue to repair the damage as best it can. You can help that tissue remodel better by doing gentle, isometric movement with the joint in question after those initial days.
You hold the joint in place, using your hand, part of your body, furniture, or a wall and press against it - for the smallest joints - about the force of lifting a coin, for the largest joints, at most the force of scooting a kitten.
As pain decreases and your range of motion returns you can increase the force until you're up to full again.
Edited to add: of course, if you have long standing joint, ligament, or bone conditions, go see your doctor! Sprains are much more complicated when stuff like that comes up.