I'm in a similar situation - 2nd marriage and all the GC are "mine" from my first marriage. He has 3 AC but they have no children.
I spend part of every day with my sons's children, taking them to and from school . I also spend Fridays with my daughter's children after school and in the evening. During the school holidays we often have grandchildren here all the time. I love every minute of it.
This has been my life for the last 3 years, since my son became a single father. Before that DH and I had 10 years together , blissfully retired, travelling, doing all the nice things we'd dreamed of . At first when the "child care " started he was absolutely aghast....I'd say that we came close to splitting up at that point. With three sons living 1,000km away and no GC he just wasn't prepared for any of this, and he didn't want to share our time together with these little people.
I handled his sulking very bluntly - I told him that these children are his grandchildren as much as they are mine, and that if he wanted to be with me he'd better pull his finger out and start acting like a grandfather instead of a sulky child. I made it very clear that if it came to a choice between my blood family and him, I'd leave in a minute. I told him to get off his computer and start actually doing things with them, "or else". He did, and now 3 years down the track he is a much loved granddad who spends almost as much time with them as I do.
If you let your DH sit on his computer / go off to the football and ignore the GC, he'll keep on doing it. Men don't change unless they have a reason to - stop giving him the choice of whether he joins in or not. Otherwise years will go by and he'll stay the same. Don't let him get away with it or you'll drift apart . Good luck to you.