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.