There are two full service properties, the downtown property and the hotel at the airport. Both are not "near" A&T university, but Greensboro is a small city (as far as cities in the world go) and although I like the one downtown, either is fine.
There are a few courtyards/fairfield inns to choose from as well.
The downtown hotel is kind of dead on the weekends, but can be busy if there is a convention in town.
Billy