In the nicest possible way: grow up!
Imagine you had a child, entered a new relationship and your new partner behaved and felt like you do.
What you are feeling is burnout. You give, give, give, but don’t get anything meaningful back. It’s not those kids’ fault, it’s yours and your DP’s fault.
So, taking the kids out of the equation for a moment, your DP seems to be using you more than you want him to. Driving him everywhere, paying for him more than you are happy to, not reciprocating your affection and bids for time.
All you have to do is stop. Stop giving him lifts, ask him to sort different arrangements for picking up/dropping off his own children. You are a new mum, you’re tired, you need your weekends. Yes, he might not like it, but you need to have some time for yourself too. So ask him to make a different arrangement, one that doesn’t involve you.
Have a good look at your blended family dynamic. Two different age groups children, they want to do different things. Find a common denominator for you all. Walks in nature and meals shared together are usually suitable for everyone.
So, say, Saturday evening is family time/meal. If you resent having to cook and he’s not pulling his weight, choose something easy: bung some pizzas in the oven, garlic bread and a big salad.
Let him parent his own children. Don’t get up with them in the morning just because you’re up with the baby. Use their snuggle time wisely: put the baby nearby/in his arms, let them watch a movie, you go have a bath in peace, go for a 20 min run, leave them to it.
Nap time for the baby: guys, the cinema, the park, visit grannie etc.
Christmas: get some frozen croissants, bung them in the oven when they get up, orange juice, let them open the presents, take photos, enjoy all their happiness. Everybody takes turns opening one present at a time. You’re absolutely within your rights to take the baby to your family for a bit, he can go to his, you all meet up in the evening.
Again, the problem is not the children, the problem is tou teo adults.