I think educating yourselves on the differences will be useful, but there is no way that it is feasible to work through all the delta files and check all of your sources for discrepancies.
Hopefully you have unit tests for your software suite. If not, then understand now why they are useful and write comprehensive tests for everything that you have.
Make a fork in your version control system, and add use strict
and use warnings 'all'
to the top of every source file if they aren't already there. Run your test suite until everything still compiles and works perfectly under Perl v5.5.
Install Perl v5.24 on a test system and adjust your sources until they pass every test.
The Perl 5 crew have been excellent about keeping backward compatibility, but there will almost certainly be some casualties.
When you're confident, make the new version live. There will be more failures, and for every one you must write a new test into your test suites.