Playing with Newborn

This might be a stupid question, but how do you guys play with your newborns?
My problem is that he sleeps, then wakes up to eat (that part is good!) however, then he is either sleepy again, or he's awake and wants to snuggle, but he has a full belly! If I put him on his back in his activity gym, he gets excited, but then he throws up and/or cries because his food is coming up. If I do tummy time, he also throws up! He does best vomit/gas wise to be held vertically after eating, but by the time 30 minutes - an hour has passed and I feel like I can put him down to play, he's tired again!
I try to get him to play when he wakes up before he eats, but he wants none of that! He's a happy, healthy baby, I just can't figure out what to do with him since he's always full or asleep! 😝