battery under 13 .4 volts, starter won't engage
 

Hi everyone! I have a 1988 300 S It starts the first time, but not the second. Has a new battery, alternator is working. After I turn it off, my battery tests out at 12.25, I charge it to 13.4 and poof! It starts. It will jump start too. Do I need to replace my starter? What's happening here?! When I go to start it after it is warm, nothing, no click. Lights are all on but nobody's home! HELP!

I tested the amps when I go to start it, once it was 600 when it started and the next time 100, the starter wouldn't engage ?!