Battery light on

Battery light comes on and off, sometimes stays off for 20 minutes, so what happened is Saturday I got in my vehicle, A/C fan was going from high to low on itself, I looked at voltmeter on dash and it was moving from almost 14 back to 12 when fan when go high to low. So, I went to AutoZone and had check engine light on, OBD port wasn't working. Changed a 20-amp fuse on the side, tried to start vehicle had to jump it, they said it was the alternator. I put a new one on 2 weeks ago from Advanced Auto, so I said crap let me try to run down to advance which was a 1/4 mile away, they said it was a drained battery, so I bought a new battery, put it in at advance and started the vehicle and battery light came on and read not charging. So, the voltmeter reads 12.2 on battery, when I crank the vehicle and put voltmeter on the battery it reads, 11.4 amps.