I have a 2007 Civic SI.
Yesterday electrical things slowly started dying (ac fan, radio, power steering, etc..) till the the engine just stopped, I was in a parking lot.
The battery was original Honda one. Got a new battery and everything is running but I'm not sure if the alternator is good or not. Maybe I'm just running on battery now. Have only driven it for a few miles since battery replacement.

Checked the voltage on the battery with engine-off = 12.5 volts.
Started the car, no lights, no AC running = voltage crept up to ~13.5 volts.
Turned on headlights and AC = voltage dropped to ~12.3 but never recovered back up to 13.5+.

Do I have a faulty alternator?
Can alternators provide power but not enough? I thought they are either dead or alive, this one seems to be in between....?

Thank you,

Boris.