It used to be easier didn't it, when our/my generation was kept in hospital for a week with a first baby? Hopefully we were more refreshed and confident after the birth, as we got much more sleep, and advice about feeding, bathing etc was always on hand. Visitors only at set times, but they usually got a cuddle with the baby. So all family and friends had a good, but time limited, share of the baby. Your new little family, no doubt, got sent home asap to cope as best they could. Your OH is right, it's very early days and they're going to be exhausted, working hard and worrying. They can't tell you that every night in a text!
I think grandtanteJE65's advice is spot on, including the lovely text she suggested.
Offering 'help' is a bit vague so why not call around occasionally (def not more than once a week) in passing, as I get the impression that you're not too far away, to drop off a home bake (or even shop bought) of something you know DiL in particular likes. Could you buy, make or knit a little item of clothing you think she would like for GS? Or take a gift token for her favourite children's shop? Maybe some flowers if she likes them. Not too much, don't overwhelm them. If you do call though, do not stay for a coffee etc even if asked. If asked, have a peek at the baby and go. If you're going to the supermarket, ask if there is anything they want dropping off. Wait to be specifically invited to go around and stay longer as they might feel they have to tidy up a bit when they have 'visitors'. Then they might appreciate a quiet hour while you take baby out in the pram. As others have said, when they are feeling a bit more confident, she'll probably feel she wants to try to manage without her DM a bit more and things could become easier for you. Do invite them around to see you on occasion then and make your home as relaxed and child-friendly as possible.
I don't agree with having a quiet word with your son though at this stage, that would certainly have annoyed me if my MiL had done that. She never did, and she never travelled to visit us btw, but they were so lovely that we visited them every week. Bide your time, have patience, and congratulations!