Cash is taken everywhere in the United States.
You do need a debit card to pay the deposit to hold your reservation, at least that is my understanding. I have never had them charge my card without my knowledge or consent. I would call them to confirm.
You need a CC to hold the reservation and they tell you they won't run it until you get there and you can choose another card if you like. Then they go ahead and run the CC they just told you they wouldn't run. If you want to pay cash I would try to arrange that ahead of time maybe with a credit card that only has a tiny bit of room on it.
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