I've moved twice when heavily pregnant and again when DD2 was a few months old. (Hmmm... there's a pattern here! I love this house, maybe no more kids!!) I found moving when pg a lot easier, particularly if you have plenty of support. I suffer SPD when pg so I was pretty useless physically but I could still stand around and boss everyone else! I really enjoyed the nesting and setting up of a new place and it certainly gave me plenty to do in those last few weeks of pg which just seem to crawl by.

Perhaps you could ask your current landlords for some rent relief to put towards the cost of removalists?

Good luck, I hope you find a lovely new home.