There are lots of places listed for anywhere from 1 night to several weeks.
And, yes, rents have gone up and the competition is fierce. With an income of $81K a year, you might be able to buy something for a comparable monthly payment.
Good luck, Jodi Selene
Here's a link to what you can find in Berkeley
This may be your best bet so that you aren't stressed out more than you are already when you are relocating.