Supposedly a fully charged unsealed lead acid battery with no surface charge should read 12.76/2 = 6.38v and can be charged at 13.3/2 V(hot ambient) to 15.5/2 V (cold ambient), but use a current limit.
Charging at a C/20 rate, 1A, shouldn't hurt anything.
Check the battery!
12Ah at 20 hrs = 12/20 A, so at 6v you need a 120/12 = 10 ohm, 7w load resistor. A 100w 120v lamp coming in at a cold resistance of 10 to 14 ohms should do it.
The "cutoff voltage" may be 5.5v.
Since your load is not a constant 600 mA current you may have to do some math.