How are you turning it off? Do you use the button on the top of the phone or do you use the option in the menu? If you use the option in the menu to turn it off, then it sounds like the button itself may be broken. If you feel brave and you have a T6 screwdriver, you could take it apart and see if the button has just lost contact. I also wouldn't give the seller good feedback for selling junk that can't even start normally, that should have been disclosed in the auction.
T-Mobile Bold 9780