House battery while hooked up

We spent about 12 days at Fort Wilderness in the Florida July summer heat.  I checked the battery after about 5 days, and once again found it "boiled" almost dry.  Refilled and 3 days later it needed water again.

With my analog meter, I could see that it only has about 12V while being charged.  I'm wondering if that means it's on its way out, assuming that the regulator would be trying to charge it at about 13.8 VDC and that the battery won't accept the charge.  That would explain why it bubbles away constantly, never getting up to full charge.  The regulator also seems to be very warm all the time, further evidence that it's working hard.

Our previous trip (Memorial Day) resulted in an unscheduled delay of 1-2 hours in a parking lot with the refrigerator running on +12V.  That weekend we had no hookups and the house battery died very quickly.  We could not revive it until we returned home to a 120V hookup.  After charging, the battery seems to discharge much more quickly than it did last year.  I did remove it from the trailer and store it in our unheated garage over the winter.  I trickle charged it once over the winter.

Thoughts or similar experiences?