on the way home from work when i started my car i noticed the battery light was on. i turned off the radio and everything to get home and made it without the car dieing. the car was running low rpmsbut still running. about half way home the battery light turned off the the rpms went back too normal, and once this happened i noticed a elictric burning smell, that consisted with the car running. tyhe smell is very faint but when you open the hood you can smell it. to see if it was my alternator i disconnected the battery before i turned the car off and the car died right away. i quickly took it to autozone anfd they said my battery was bad whitch made no sence because the car still startes and drives. its only when i disconnect my battery while the car is running is when it dies. the battery light has not come back on but i can stiull smell the burning smell.. i dont wanna spend money on an alternator if it could be something else. PLEASE any advise will help. (the lady at autozone didnt seem like she really knew what she was talking about)..

01 grand prix gt 3.8 146K (original alternator)
sorry for the novel(i only drove the car about 20 since the battery light came on then went off. what to do)??