There are at least hundreds and probably thousands of locations. AAA is shockingly good. Maybe not shocking if you consider it's a 120-year-old nonprofit member services organization, but still: anyone who drives an auto in America is leaving serious value on the table if they aren't a member.
The bottom line is that if you're having a problem and you're in an automobile (doesn't have to be yours) AAA will do their best to help you solve that problem.
Unlimited, free, high-quality paper maps are just another perk. Walking in to a member branch and walking out with maps is just the beginning: a AAA employee will help you plan out a road trip, and make what's called a TripTik, which is a custom spiral-bound route map, with various sorts of amenities you can choose pointed out for you.
There are campgrounds as well. It's truly remarkable how much AAA offers.
They are very common--even tiny cities in the US will have an office somewhere. In fact, they're one of the few places to easily pick up properly formatted and valid international drivers licenses before you go overseas.
However, I would caution you that some of the benefits that used to come from being an AAA member have been severely curtailed. The towing benefit, in particular, now has quite a few restrictions on it.
(Because, in Germany, even if you're an ADAC member, you'd be hard pressed to find an ADAC-affiliated office to pick up a map from...)