First thoughts: Could have been a bad transistor to begin with, then it became a coil diode issue (back EMF). The wiring to the coil terminals got mixed up, or the diode polarity was mixed up (coil was recently replaced).
Other thoughts: Perhaps too much current being drawn from a short in the wiring or in the coil winding. Is the correct solenoid circuit fuse in place (why wouldn't the fuse blow)? Was the game playing ok with the new transistor, up until it failed. Miss-marked or re-marked transistor, try another one. Something upstream of that transistor is locking it on.