This is going to sound odd...but I had pretty much the same thing happen two weeks ago on the second day out on a 17 day sailing adventure. We motored quite a bit on the second day and I noticed the the batteries were showing 12.4 volts several hours into the trip, I immediately powered the motor off and inspected the wiring suspecting a loose connection, turns out that everything was tight....or so it seemed.
At the end of the day I did a really deep dive and finally found that the positive post on the alternator itself was loose....inside the alternator. I could start the engine and when I reached in and pushed the post ever so slightly toward the side of the alternator it would move and the motor immediately loaded up, remove the finger and it would unload.
I removed and disassembled the alternator and found the two nuts that tighten the post to the rectifier had backed off causing a loose connection. There had been a good bit of arcing going on in there and the back of the bottom nut was pretty much gone, the rectifier itself also had a bit of damage to it as does the positive post itself.
I was able to clean things up and tighten the post down to the rectifier and got everything working again but I'll definitely need to go thru the entire alternator and replace a few parts.
The moral of this story is....don't just check the connections, check every component you can and suspect everything. The alternator was brand new just two years ago so it never rose to the surface on my suspect list....although it was in fact the problem to begin with.