Hi Pisces here that has moved 10xs in the last 6/7 yrs. I am an expert at this.
Step one is pack up all the non essentials a good 1-2weeks before your move date. This also gives you the opportunity to sort through things and create a goodwill pile.
Always overlap your rentals by at least 1 week, I've always done 2-3 weekends at least. When you get the keys for the new place bring cleaning supplies and do a deep clean right then and there.
You won't need all that many boxes, just what fills up your vehicle as I am a big believer in unpacking right away. So each round of moving you unpack and then take the boxes back for the next round.
First weekend = small stuff/boxed unessential's. Also bring a blow up mattress so you can crash at both places if needed. 2nd weekend = furniture move... bring the friends with trucks to help you move all the big stuff you can't handle alone. Since new place is already clean you can setup everything right away. 3rd weekend = deep clean on the old place and walk thru with landlord and get his signature before you go. Deposit is gonna be secured no matter what!
This method has served me well because during the week I am too slammed with work to take time off and move. Also fairly fragile when it comes to my living space and need a peaceful clean environment to be productive. So having some extra time keeps the pressure off and allows for an easier transition.