All of our moves have been without pressure of a time frame.
I agree with doing all the work you want to do immediately BEFORE you move in. SO. MUCH. Easier. And paint your heart out. DO NOT leave any painting to after the move in. Truth is once you move in any future paint jobs will never happen.
Said from far too much experience.
But truthfully once we got the house ready for move in, the actual move in has never been drawn out too much. It's VERY hard to live between houses. What you need is ALWAYS at the other house. So honestly once we start we just plow through to the end.
How far is the new house from the old house?
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As to your specific questions. Yes. If budget doesn't allow for anything else, definitely do a deep clean of every surface before moving in.
My 2nd priority would be painting as much as possible.
When you say rugs, do you mean floor to floor carpeting? If so and if possible it's much easier to replace carpet in an empty house BUT if the budget doesn't allow it before move in I'd clean (if you have access to a steam cleaner) and deal with it until budget does.
For the actual move in, I did a room at a time. Easy to pack up on area and drive it over and immediately put it in place (since I had the luxury of a local move).