Hi mlsunshine! I think having the kids fall asleep in one room while being looked after by one set of adults is a great idea. We did something similar years ago when we sailed with a family group. My sister and I took turns watching each others' kids. Of course, you can certainly bring them to the show lounge if you think they'd enjoy it--and if it's not the late night adult show. We sat in the back with our son when he was younger in case we needed to make a quick escape, and he usually enjoyed the shows.
On another subject, I highly suggest that you bring a note from your OB/GYN/midwife indicating your due date and your fitness to travel. Princess does not allow anyone beyond the 23rd/24th week of pregnancy to board, so you need to be prepared in case they ask. I'd advise checking with your TA regarding Princess' rules regarding any official documentation that may be needed.