Perhaps what you heard might be in regard to photo licenses? You do need an official photo ID to rent a car in the US, and older UK licenses didn't have a photo. However, as you do have your passport (having just entered the country, after all) that can take care of the photo aspect of the driver's license requirement.
BTW, if your license does not have a photo yet, you should carry your passport at any time that you might wish to buy an alcoholic beverage; without a photo ID that shows your age, you will not be served.