I think you know your own intentions here, this is not just that you want this man to carry on being a good Father, he can do that without taking you along on trips and doing jobs in your house for you. It sounds like you're still in love with this man and can't accept that he is now in love with another woman.
I''m in my late 40's now but spent much of my 30's and 40's as a single Mum. I didn't have a new partner for a number of years. Even so, I would never have asked my ex to come round and do jobs for me. If anything I wanted him to realise that I was an independent woman who didn't need him for those things. I learnt to do almost everything in my home for myself (although any kind of plumbing is my achilles' heel!
My ex would pick up our children on the weekend, take them on trips and holidays by himself, and when he got a new partner (who is a wonderful, kindhearted woman) she would go along too, I would stay home and enjoy the free time which is what I think you should be doing (you're here for opinions, therefore I'm giving mine). This is the natural progression, you are no longer his partner, she is, you are prolonging your agony with this behaviour.
You really need to move on from this situation, it will mess up your mental health as he becomes less involved with you and more involved with the new partner. How do you see the picture if and when his new partner has a child with him? Do you still think she should come second to you and your children?
You are definitely being unreasonable with your demands on his time and attention. I do understand it, unrequited love is one of the most difficult things a person ever has to deal with, it is a bash to the ego and self esteem and can take a hell of a long time to get over. This man ended the relationship with you for a reason, and it doesn't sound like he's coming back. You need to focus your attention elsewhere and crack on with your own life.. without him x