Use dollars to make small purchases and a major credit card to make larger purchases. I the two places you are going the US dollar is so readily accepted you virtually see nothing quoted in local currency.
Please know that the American dollar is only worth 80 cents on Grand Caymen, which is going to make anything you buy there cost 20 per cent more. Even a taxi cab ride! And they also automatically tack on a 15 per cent gratuity to all restaurant tabs, so you end up paying a total of 35 per cent more for anything you eat or drink there!! Needless to say, we stay on ship whenever we hit that island on a cruise.