how old is your battery? if it is over a year old it is possible that it could have a dead cell. that means it will read like that battery is good but it will not hold a charge. before you tear out starters try swappin in another battery and see what happens.
Hey i had this same exact problem, my starter even worked i had it check out. my battery was fulyl charged too. I check the clutch switch since i have a manuel and it still worked, from what i was told it sum safety switch sumwhere on ur transmission that has gone out. Thats what gm told me, i just ran a push button start on my starter, cuz i couldn't find the switch, and it was a lot cheaper. I light it too.