alternator question

(2011 corolla 1.8l) hey everyone, earlier today when i was driving thru the city, my battery light suddenly kept flashing. i turned around brought it back to my house so i could test and see if my alternator was going out. the battery was at 12.8v with the car off after a couple of hours and got up to 14.1v with it idling with no accessories, and down to 13.89v with headlights, ac and radio on. i took it for a drive to see if i could get the light to come back on, and after 30 mins of city and neighborhood driving i couldn’t get it on. i tested at home and the battery with the car still running was at 13.3v and very slowly climbing. my question is, is my alternator going out? since my light was flashing i would think so but 14.1 at idle and 13.8 at moderate load seems like at the very least its still managing a usable charge. and yet, my light was flashing? should i go ahead and replace the alternator or maybe direct my attention towards something else? thank you all.

also, the battery is about a week or two old, if that matters.