Those V readings look very similar to what my old 130k oem unit put out with a bad set of diodes. I replaced it (Still worked) and with load/no load it'll never go below 13.8~. I'm pretty sure I took pics of the voltage readings on both alts before and after. I'd have to check when I get home.
Is it failing to start after sitting a day or two? That would indicate a possible parasitic drain somewhere in the system. if it's just a lower than optimal reading and the battery tests ok then it's likely a weak alt that will in time, if it hasn't already, kill/severely weaken the battery.
If you're driving short distances with a load it's very possible that the alternator isn't putting out enough to charge the battery properly. As it sits, it's pretty much outputting standing battery voltage while under load.