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.
That is a pretty good image from an old folder. You need to stop that down to get the edges sharper (if that is what you want). One side is worse than the other because when it unfolds, the lens and film are not parallel. You can try to fiddle with it, but there is no comparison with a Rolleiflex TLR, with its rigid box-like construction that keeps lens and film parallel. Even if you did get it aligned, the lens is still not very sharp at the edges. I'd just use it as it is and get a better camera if you want sharp corners.
Could also be an issue with film flatness on the scanner if you are scanning.
It's possible the camera was dropped to a carpeted floor at some point, knocking the lens standard out of square ever so slightly. And with a carpeted floor, there wouldn't be obvious signs of damage.
However, I agree with others to first make sure the pressure plate has enough pressure to keep the film flat. The waxed paper idea across the film plane is a good one, too.
Once again thank you all for the solid advice. On checking the pressure plate I noticed it was not lying parallel to edges of the "door". Manipulation caused the leaf springs to free up in their slots and the plate looks in a better position. I will also work on the leaf springs controlling the film roll.
This is a great site. Thanks.
Folks, it WAS the back plate. See the two attachments below. Left is after back plate adjustment. Right is before adjustment.