I would gladly upgrade to a more recent Perl, but that may break some of the programs.

Perl rarely breaks old code. That's why modern Perl needs use strict, use warnings, use feature - changed defaults would break old code.

We don't have the Perl programming expertise at our organization to handle that quickly - and we need to handle this quickly because our Win 2008 server contract ends 1/14/2020.

Roll out the changes (i.e. the new Perl version) to your test environment and check what breaks. No need for guessing and prejudices.

No test env? Time to clone your production env ASAP. Having a verified(!!!) backup of your production env also sounds like a good idea.

Alexander