A similar problem hit my phone late last week; it would detect the USB charger only if the phone was off. I do not know if using a wall charger instead of off the computer would have changed anything.
I wound up removing some recent apps, restarting the computer, and pulling the battery. One of those things made it work again.
Were it my phone with your symptoms, I would try charging it with the wall charger. If it was still going through the charging->starting->not-charging->turning-off steps, I would take off the back cover and once the phone turned itself on, pull the battery so that the phone crashes instead of sleeps. That way the phone won't try to turn on when you go to charge it again. But I give no guarantee that that will work (or even that it will not damage your phone, it's just what I would try).
EDIT; The real first step would be to borrow a charged battery from someone else, start the phone and turn it off. But, I don't know anyone with a Droid, so I never think of that first.