Looks like a mini lathe of some marque.
Have you checked that the (probably plastic) gears are: 1) Intact – No broken teeth or torn keyway)
2) That the keys are present in all the gears from Mandrel to the Leadscrew, (otherwise you are relying solely upon friction to provide to drive to either the Leadscrew, or the pinion and the Saddle)
If the geartrain includes a compound gear as an Idler, there should be a key long enough to unite both gears, and similarly, the key on the Leadscrew should be long enough for both the gear on the leadscrew and the spacer
Without the spacer, the gear could float and not drive continuously. The gears should mesh over their full width.
With regard to ,meshing, there should be just enough backlash on each mesh to allow a strip of paper to pass through, no more and no less. (The Banjo and the Idler stud(s) are all clamped up properly, are they?)
One of the gears is not running eccentric is it?
DAMAGE reported: ANYTHING bent, Leadscrew, shaft in the Saddle, (might account for the "slipping on the rack"
Not the tumbler reverse?
As suggested, maybe you need to ask if anyone local to you would be preared to come and have a look to see where the problem lies, and what might be done to solve it.
Howard