Congrats on getting the move part behind you. It won't be long now and the settling in part will be behind you as well.
As for the ruts... surprisingly easy to deal with if you're so inclined.
Using a round point shovel like this one:
Eagle 46 in. Long Handle Steel Round Point Shovel-1554300 - The Home Depot
You just need to lift the rut above grade to eliminate the soil compaction and nature will do it's thing with future rains to return the soil density back to normal.
This process goes quicker than you'd think.
You put the shovel nose in the rut vertically as deep as you can and pull back on the handle to lift the sod. Move back 8-10 inches and repeat. Don't pop it up too much above grade or it'll cause issues with mowing. I was able to repair ruts left on my property by a well drilling rig in pretty short order. Obviously it'll help if the lawn has seen some rain to soften things up and not knowing if there's a lot of rocks or roots at shovel depth but that could slow things down as well. I didn't have anything like that under my sod so I was done way faster than I thought I'd be. The drill rig guy showed me how to do it and I was grateful.