I did a lot of boat work before the depression and I often saw this sort of thing.
Assuming you have a good meter for starters, mine are. It is not uncommon to see a fully charged battery at rest sitting at 13V+. With even the slightest load, the terminal will drop into the high 12s instantly. Turn on a light and they will be a 12.6 in moments. I was told by my "battery guy" (an engineer with Decca) that this is an artifact of "surface charging" where the electrolyte nearest the plates has developed a higher density and has not been redistributed yet. That redistribution can take quite a while.
The condition does not represent or cause a problem.
Matt