I haven't got a glue gun, so I'll give the tape a try.

Do the glue. Hot glue guns are the second best invention in the world next to the empeg, you should definitely own one just for the sake of owning one. And as far as "how to do it" goes, the Drive Upgrade Guide has a very clear video of exactly how to do it. I highly recommend you look at it.

Another thing to consider is the way the drive cable is folded and positioned inside the player. If it's being pulled loose all the time, maybe it simply needs to be creased a little differently so it doesn't constantly have that pulling stress on it.