Yea I absolutely get the concern.
If I may offer a suggestion, if you are offered a spot without ample time to move-
At the end of August/start of September we get a long weekend with Labor Day off. And as our exams are nearly always on Mondays, that makes it a week without exams.
Totally not ideal but if you needed to make more than one trip to move all of your things down, I would keep that weekend in mind. If you have a spouse or another adult moving with you, you could also stagger the move and have them arrive that weekend instead of at the same time as you.
Hopefully it goes smoothly and you do not need to make plans like that, but just thought I would throw that info out there