Any moisture left on the rollers/cutters is an issue in my experience, and if left can result in pitting or corrosion either of which will be difficult to overcome. Qil I would think would not lead to corrosion, but might pit. While present, it will definitely increase the tendency to stick in my experience.
With the rollers, I would dust it heavily including giving it a couple of cranks, then run some fairly dry dough through repeatedly to grab any liquid or oil it can. Make sure to get all areas of the roller edge to edge. I have had luck with this, particularly when my roller was new and for some reason they put some type of dry lubricant on that marked my pasta grey/black if it ran to the edge of the roller. The same can be done if you do things like make flavored pasta or dough and think some residue may be left that will effect future projects for instance I make some cookies with a sour cream dough which are much easier to make with my pasta roller but I want to make sure no dairy is left on the rollers after so this is a cleanup run of some cheap four dough I will then discard.
The same should work with the cutters as long as they have not been pitted or corroded. For them, I would suggest using the same dough, towards the dry and still side, and thick enough to get all areas of the cutter that might be holding residue.
Hopefully you can get it back to good behavior with this type of dough cleaning. It happens to be the cleaning method that came recommended with my roller. When done, dust the maker and put it away that way, yes, looking "dirty" with a dusting of flour to protect it from moisture getting on it and pitting or corroding the working surfaces. Yes, some of that sounds contrary to normal kitchen cleaning practices, but it is what was recommended by my manufacturer and has worked well for me. It is clean, dry flour you are putting on which would not seem like a major safety violation, just a little unsightly.
PS, if you do not want to waste eggs, and I do not blame you, just mix up a dough with a bit of flour and water. I still hate wasting any food, but at least flour is one of the cheapest things we use, and you do not need much. Far less loss than needing to replace the entire pasta maker set.