I am not sure that GN is the right place for best advice, but if you were on MN I am 100% certain that you would be told "GET RID" in no uncertain terms!
I have been married for 53 years and my DH is very much like yours. He says he will do "things" and rarely, if ever, does. He says he forgets, hasn't had time, will do it tomorrow/next week/when he has time (which apparently he never has). However, it is only recently he has become angry or annoyed with me for "reminding him when he doesn't need reminding" while in your case it seems to be very early in your relationship. I do know that his reluctance has caused me a lot of distress as it seems that we have never shared the same aims or wants in life, and I am worn out with having to do everything in the home if I want it done. That said, he is usually happy with having it done for him!
But it doesn't sound like a fair partnership to me
I don't really have a solution for you (if I did I wouldn't still be waiting for my DH to actually DO what he says he will), but I imagine there will be one of two outcomes - either he changes his approach to sharing a life with you and your child or you accept that you are maybe going to struggle like this for a very long time.
One other thought is that he could have ADHD. People with it do procrastinate a lot, get distracted from what needs doing, take a long time to complete tasks etc. One of my SILs is like that - he means well, but my DD has learnt ways of guiding him to do things in small chunks rather than facing him with what he sees as a mega burden. But he has never been nasty to her and realises that he needs to find ways of dealing with the ADHD for both their benefits.
Good luck.