My preference is to annotate exceptions to what I am signing in legible block print, and then initial and date at the end of each annotation.

When I sign, I include my full name and the date in block print as well as "with exceptions."

Bureaucrat/legal types usually understand that asking me to sign a fresh copy without those annotations is explicitly asking me to lie.

I don’t know about that. One requirement of a contract is a “meeting of the minds,” i.e. both sides are in agreement. An X indicates that, but “unverifiable” would seem not to.