Sleeping in the car is not a good idea. I would go over to Walmart and buy basic camping gear. When the trip is done, give it to the folks that were kind enough to loan you the car - or donate it to Goodwill/Salvation Army.

Traveler's checks have fallen into disfavor over here with the transition to a "plastic" society. Probably the easiest thing to do is use your UK credit (not debit) card (Visa or Mastercard), when you need cash, go to any bank and get a cash advance (there will be a fee, and your bank will probably start charging interest immediately). The only downside to this is you will probably have to go inside to pay for fuel, as the pump reader probably won't accept your foreign card (but the inside readers should). They are set up for a 5 digit ZIP code verification, and debit cards use a 4 digit PIN. No RFID chip verification here yet.

It wouldn't hurt to also get a prepaid Visa card over here to use where your foreign card may not be accepted. There is a fee involved to purchase and reload them.