something is either up with your mounts or your optics. Maybe a deformity of the mount or it isnt settling down apropriatly. my may want to try to continue to shim with a trimmed piece of soda can, but that will put stress on the scope tube. I would change it out with another scope, if it then zeros better, you know its the scope, if not... then you know it is at least the mounts, maybe the scope and mounts.
hopefully it isnt somethign like a poorly machined action.

"several cheap scopes"
what is a cheap scope to you? are we talking NCstar/bsa cheap, or bushnell cheap?
since you have several, you could sell some of them and scrounge up the change for a nicer scope and mounts.