You will always do best using pesos, but US dollars are widely accepted. If you pay in US dollars you’ll most likely get your change in pesos. In town you will see several money exchange locations (casas de cambio). They post their rates and will exchange US currency to pesos.
ATM’s are easy to find and will dispense pesos. Some banks do charge you currency exchange fees and/or service fees for using ATM’s in Cozumel. Check with your bank in advance.
Credit cards are not accepted at all stores or restaurants.