So you are roughly spending the same as him towards the household costs BUT he earns more and apparently doesn't recognise your contribution in terms of caring for the children, running the home and making sure bills are paid.
If you do get those extra hours could you get childcare in place? Possibly with help for that?
As for it being "too expensive" to do the travel weekly for him to have contact with the kids why would that cost be solely yours? Personally I think it would be on him to cover - half of it at the very least!
It's utterly ridiculous that not only is he spending so much solely on himself he's expecting you to bail him out when he fucks up!
I'm afraid I definitely agree you should put the savings in your name only, stop bailing him out and seriously consider leaving him.
I've been a single mum 16 years, it's not easy but is possible and I can't help but think that you'd likely actually be - certainly long term - much better off financially without him AND better off mentally/emotionally.
You're both the same age, he has kids time he grew the fuck up and started taking responsibility!
At the very least I'd be making an appointment with a money management charity, expecting him to also attend and maybe THEY (I suspect particularly if it were a man the appointment was with!) can get through his thick skull that parents don't get to spend freely without FIRST making sure their families needs are covered appropriately.
What the hell is he spending all that money on anyway?!