However, it tells me that @INC contains 5.10.0 directories instead of 5.12.3

Based on the additional info provided, the 5.10.0 directories are just additional directories to search. That means that pure Perl modules that were installed with 5.10.0 don't need to be reinstalled. Modules installed using 5.12 will not go into those directories. (They'll go into /usr/local/share/perl5.)