code executes, but none of the rows are updated.

As people have suggested individually, one of two things is going on, and I can't tell whether you've provided enough information for us to tell which.

If the schema (as show by describe mytable) shows that unsubscribedby can be NULL, and if that column holds NULL (as opposed to an empty string), then

    where unsubscribedby=''
[download]

won't match. You'll need to write

    where unsubscribed is null
[download]

instead.

But, if that column really is holding a blank string, then changes are good that $_ doesn't hold what you think it does. Perhaps leading or trailing whitespace, or a newline on the end.