Have you tried replacing the battery? Seriously, thats sounds like the issue in my opinion. Think about it, you can charge it and it works for a short duration but then goes completely dead and when you place it back on the charger you can power it back on again. Your assumption that because when you take it off the charger it shows full battery means nothing. There is a BIG difference from the battery having full voltage (showing a full battery meter bar) and full amperage (being able to actually power and use the device for X amount of time).
I really do not see it being a ROM issue. If the battery contact pins were no good then you never would be able to charge the phone and much lass turn it on while on the charger because the battery has to be in the phone (ie making contact with the pins to main board) to turn it on.
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Loving my rooted Droid X
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