I am not familiar with the Perkeo, but on other 120 folders the film runs over a set of needle rollers on either side of the film frame: keeping the film tight against these rollers is a large part of what keeps the film flat. There is usually a leaf spring that rubs against the film feed roll to provide tension as the film is wound on. If the spring is weak or missing, or if one of the rollers has gone missing, then you might see the problems you have described.
Another common problem area are the leaf springs at the back of the pressure plate. They are usually slotted and slip under the heads of a set of rivets. If a spring has come unsprung then the pressure plate won't work correctly. Also, if these springs have become weak there can be problems.
Often bending the spring back a little bit will restore things to good order.