Make a Plan
It sounds pretty simple, but last time I moved, I totally forgot to cancel my renters insurance in advance, and ended up paying an extra month's worth of coverage for an apartment I didn't even live in anymore!
Also, having a plan means you can coordinate your work schedule with when you're moving, or giving a friend a specific time to come over and help carry that heavy furniture.
Give Yourself More Time
Another way to give yourself more time is to plan for your leases to overlap by more than one day. This means you can get a feel for the space, and have more time to move things (since it's going to take longer than planned). I highly recommend this for a move in town (obviously if you're moving across the country, it doesn't quite work the same way).
Offload Stuff in Advance
Chris has sold quite a bit of excess furniture on Craigslist and Ebay, and I went through my closet and posted everything I wanted to get rid of on Poshmark which is an awesome app (and site) that is for selling gently used women's clothing, jewelry, etc. (BONUS: Get $5 for signing up with the code- BLWZH)
As I'm packing (in advance, go me) I've been throwing things out and adding to a growing pile of things to go to Goodwill. You could do this on the other end of the move, but why move four boxes worth of stuff when you could move three?
I'll let you know how well everything actually goes with my stress level, so wish me luck!