Okeh. A fully-charged "12 volt" battery, once the surface charge is skimmed off by running the lights for 3 minutes, should show 12.6v. I was wondering if you were going to see something like mid-high 10s, which would indicate a dead/dying cell in the battery, which could cause the symptoms you're seeing. But no, this what you've measured tells us all six cells are at least more-or-less alive. Other battery tests (such as your drop-after-charge results) might still point at a battery problem, but I don't think it's causing your poor-charge symptoms.