Thank you all for your prompt feedback. I have a couple of follow-up questions.
First, assuming the alternator is the problem, which seems to be the initial consensus here, is it possible that the alternator is providing some power but an insufficient amount to continue to run the engine, and that the batteries have been filling the gap?
Maybe. Or maybe the alternator's no longer doing anything at all.
Btw: Being as battery #1 died while the engine was running, switching to "both" was sub-optimal. You actually reduced your run-time that way, because now battery #2 was also trying to replenish battery #1.
Secondly: I wonder if it would've been a good idea to alert the CG of my situation and establish a check-in schedule, in case battery #2 bit the dust before I was able to complete my journey?
Second, assuming the alternator is the problem, for how long do you believe a fully charged battery can continue to start and run and engine (and various other electrical devices) without receiving a full or partial charge back from the alternator?
There are so many variables there, I wouldn't even try to guess a time.
Third, let's assume I troubleshoot the alternator in accordance with the guidance you all have been kind enough to provide, and I can't identify the problem, what will it cost to replace, repair and/or rebuild an alternator,
You're not going to try to repair your own alternator. Nobody does that. You're going to obtain a new or rebuilt replacement.
and can someone who is mechanically inclined but not experienced (i.e., me) install a new or repaired/rebuilt alternator? Would you recommend me doing so?
Yes and yes. It's trivial.
I recommend you head on over to Moyer Marine
and pick up the A4 Service and Overhaul manual. Indispensable.
Btw: It might be just the belt's off (broken). AutoZone carries the exact belt you need , if that's it