I just re-read your post. If you do not have full coverage on your personal vehicles, then more than likely, you do not have coverage for car rentals.
If you do not have full coverage on your own vehicles, you may not be able to pick up your rental without the purchase of insurance. I know my Dad used to keep full coverage on his vehicles (no matter how old or how little they were worth), just for this very reason -- without full coverage on your personal vehicle, they may not allow you to rent a vehicle without purchasing their insurance.
I know time is short before you leave -- but you might call your Insurance Agent, see what it would cost to add full coverage to your policy for ONE month -- get a temp card with full coverage on it -- and then you should be good to go at the car rental counter. Be sure to bring that temp card with you!