There are only three things that could cause problems with the slippage you are encountering, Geneva. The first should be easy to dispense with, that of the tape which holds the liner sheet to the takeup tube not holding on. If, when you finish laminating the liner is still taped to the takeup tube in the same place where you originally taped it, then this is not your problem.
The second would have to do with the cardboard takeup tube slipping on whatever is inside it holding it to the takeup bar. I looked at a picture of your laminator, and whatever it is is completely inside the cardboard tube, probably sliding on a hexagonal cross-section takeup bar. There should not be any slip possible between the hub and bar, so if there is slip there it is between the cardboard tube and the hubs. Our laminator has this type of hub on the bar which holds the roll of laminate, and some brands of laminate can slip on the hubs. To fix this we use a long screwdriver to scar up the inside of the cardboard tube so the hubs can grip the cardboard better. (our hubs have floating metal strips which angle to grip the inside of the cardboard tube). If your hubs have a different way of gripping the cardboard tube, you may have to devise another way of increasing the hub's friction/purchase on the cardboard.
If there is no slippage occuring outside or inside the takeup tube, the other problem could have to do with the "takeup pressure" adjusting clutch, i.e. the takeup reel just isn't pulling hard enough on the liner. If that is the case, adjusting the takeup tension knob will fix that.
