You can polish your old Lee die using a snug fitting shotgun cleaning mop and J&B Bore paste, automotive rubbing compound etc and a drill.

The scratches you are getting are from dirt and grit embedded inside the die and polishing will remove the dirt and grit.

I went to wet tumbling with stainless steel media to scrub my cases before sizing to eliminate scratching my cases.

And as others have stated above stuck case tools can be bought or made to remove stuck cases.