I had a similar issue when I replaced an APM with one that I thought I fried. The spare was from a crash of another ship that I salvaged. I reloaded the firmware, completed all the calibrations, but when I took my hexcopter out for a test flight it immediately nosed over. Two other attempts failed miserably and I decided that the APM was damaged.
I ordered a new one from Amazon and it works fine. I don't know if yours is new or not, but if you've tried everything else, you might have to replace it. One last thing - I just skimmed the posts and I may have missed this suggestion, so at the risk of repeating I'll ask:
Have you checked the Center Of Gravity (CG)? You should be able to hold it up with two fingers on either side at the middle of the chassis and it should balance. You'll have to do this as well fore and aft.