Alison, when I worked at newspapers full time I got RSI in my right elbow and shoulder (I am right-handed). All those years on the computers really took a toll on me, and I was in a lot of pain for a while.
I did go to a doctor and, like you, had NSAIDs. But no splint. (Be careful, though, they'll put you in a sling if you wait too long and overuse the joint too much. Better to go sooner rather than later.)
One thing I did was retrain myself to use my left hand as my computer mouse hand. I just decided that my right hand had borne the primary responsibility for things for too long and that my left hand was going to pick up some of the slack, lol! I also tried to realize how many things I did with my right hand to see if I could do some of them with my left hand -- carry my purse, brush my teeth, carry plates to the table, etc. It probably took me about six months before I was back to normal.
Of course you need both hands to pick up your son, but if you can lead with your left, maybe it will actually strengthen your weaker side. And perhaps there are other areas where you could retrain your left hand to do tasks and give the right shoulder a break. Things like making your left hand be the primary hand to pick up toys, clear the table, brush your hair or hold the hair dryer -- any of these things that the left hand can learn to do gives the right shoulder a break. And then perhaps there are some shoulder stretches they can give you that will help.
I'm commiserating with you. My sister has four kids, and two of them were 15 months apart. She'd probably tell you that her aches and pains didn't start going away until the youngest was 3 years old, but I don't want to discourage you, lol!