#1. Take the battery and give it an 8 hour charge, disconnect and let it sit with nothing connected for 24 hours, then go and check the resting voltage, should be 12.65V, if it's any less than that, ie, 12.4, it's only at 60-70% of charge and on its way to the recycle shop.
#2. With the voltage drop you see, remove your conectors and give them a good cleaning with a pencil eraser, if there's any amount(I mean ANY amount) of corrosion at crimp on connectors, re-wire and solder on the connectors.