You say the shaft moves back and forth. Does it move in and out, or, up and down. If it's moves in and out the problem could be too much end play. If it has too much of that maybe the fan is dragging on the housing and you could find a way to give it more room to move and then save yourself a few bucks for a few years .
If it is too much end play I have taken fan motors apart before and found a way to add a new thrust washer; made out of something; and eliminated the end play.
If you are thinking of throwing it away anyway you have nothing to lose by dissembling it.