Insurance is zip code specific - so moving from one zip code to another can cause your premium to change. Especially if you move into a more crowded urban area (like a downtown area) because there are new factors of risk your insurance company has to compensate for.
You will need a permanent address to get a policy with most companies so I would recommend using your previous address until you move to a permanent location. At that point, you can update the address on your policy.
Unfortunately it seems like you are in a tough spot until you have moved into your new place. Lots of information is used by insurance companies to determine the rate that they will charge to insure a driver and a major factor in rating that price is your home address.