Thank you Julia for your question:
Terri Vellios said it best, yes you still need to pay your rent.
Also, at the new place that you move to, your future landlord will probably ask to talk to your previous landlord. You want a good reference from the landlord where you are living now. If you stop paying your rent, I guarantee that you will not receive a good reference from your current landlord.
Right now in San Jose thee is an enormous amount of competition for the few places that are for rent. If you want to have any chance of getting another place to rent, you must get a good reference from the landlord that you have now.
Charles Butterfield MBA
Real Estate Broker/REALTOR
Office Phone: (408)269-4238
Cell Phone: (408)509-6218
Email Address: firstname.lastname@example.org