The switch worked before I added <some code which could not possibly affect the switch statement>.

This wonderful and mysterious feature of the Switch module is the reason it is so warmly deprecated. It arises from the fact that the module is a source filter.

What else could I use?

For an equivalent in 5.10+ versions, see the  given and  when keywords discussed in the section Switch statements of perlsyn.
Prior to 5.10, see the section entitled something like "Basic BLOCKs and Switch Statements" in perlsyn for alternatives.