The streaking do not 'dilute' the plasmid, but simply purify the colonies.
The copy number of the plasmid mostly depend on your replication origin (
never use 2u origin if you want a low copy), and whether the plasmid contain
centromere (like CEN6 on pRS416). Usually the centromere sequence can limit
the copy number of your plasmid.
I don't think you need to wash or replica your plates. If you don't want to
have too many plasmids in one cell, just simply reduce the transformation
efficiency and do more transformation. Or you can culture your transformants
for 1~2 generations before you plate them.
BTW, you should have some ways to select your candidates. Then culture the
cells under selective pressure should allow you to enrich the plasmid you
want.