Have you checked the voltage output of the alternator? Sounds like it's not working correctly. Let an auto parts store test it. Depending on year model the PCM controls the voltage of the alternator so it could be bad or possibly one of the connectors going into it could be tarnished or not making connection.
Sorry, title should be new alternator and new battery.
Brought it to a mechanic who installed a new battery then after install told me the alternator is no good. Problem is I had already replaced the alternator three times in two days because I had assumed they were bad alternators. The mechanic saying that about the third alternator leads me to believe it something else. I cleaned all ground connections. Something electrical? Fuse?
Gp to Harbour Freight tool and purchase a battery/alterantor checker. Check to voltage coming off the alternator, then check battery voltage. You may have a grounding problem? ground to engine connection.
Tighten everything on these connections.