Battery voltage doesn't tell you much on an old one. It's the capacity that counts, and is lost in the ageing process. It doesn't matter how long you charge it, either with a long run or on a charger, a voltmeter will show it fully charged around 12.7 volts but only on the fraction of the as new capacity remaining.
Symptoms described are a classic example of knackered battery failed as above. It's a consumable, fit a new one.
