Your post is a little bit confusing, but will try to unpick a bit:
My partner ....... obsessively compared and nitpicked about the house/area we moved to
That seems pretty sensible to me. It is a big step.
Not clear if you are renting or buying ?
Either one costs a lot to do, both in money and in time and stress. Better to get it right in the first place than be wanting to move again soon afterwards.
He is now having doubts about moving again as he now likes the area we live in and has made good links.
Well, all good reasons to stay, IMO.
My partner and I are about to move. We are moving a little closer to my family. It's not set in stone yet but i'd like to be closer to them.
This sounds to me like you mean you want to move house to be closer to them, rather than it being a joint decision.
We've had a horrible fight over him saying that, 'oh, it's such a lovely park' near where we currently live and I can just feel myself being needled.

I genuinely cannot see how this isn't one of two people making big life decisions together, expressing an opinion. Honestly, I'm not seeing why this "needles" you ??
I feel like I do most of the practical chores and organisation despite working exactly the same hours.
Completely separate issue. Absolutely it needs addressing, but don't tie it in with the house move. Discuss it as the issue that it is.
i'm a reasonably funny/fun girl and he rarely laughs with me. Everything is mega serious or unnoticed.
Again, this is about your relationship. Relationships do usually change when you move from 'dating' and being a young couple having fun, to being "grown ups" with a house and a child you are responsible for. Lots of people find it hard. Because it is hard to work and to parents at the same time. What you need to do, is talk together about what you might try, to reconnect with each other, not ask him 'why he rarely laughs with you' - as that is you blaming / accusing him, rather than you saying "I miss it when we used to {insert X/Y/Z} - how about we book a babysitter and go to ..... sometime soon"