[Mark44]

How old is your battery? I usually get 8 or 9 years out of a battery, but people who live in more extreme climates get significantly less. A reading of 12.6 volts is not very good, and 12.3 volts is very low. A chart I have puts 12.6 volts at 50% and 12.2 volts at 25%, but these percentages might be low, according to other tables I've seen.

From your readings, it appears that your alternator is working fine. You said the battery connections are tight, but are they clean? Undo them and check that the inner clamp surfaces are clean. If you don't have one, get a battery terminal cleaner. They're pretty cheap, and any parts store would carry them. Is there any corrosion showing on the battery cables? Is the grounding connection nice and clean? The ground cable wants to be bolted to shiny metal.