Perl should always search for libraries in your directories in the order that they appear in @INC. If '/path/to/my/dir' is the first thing in @INC, perl will look there first.

You might want to try something like this:

#!/usr/bin/perl
use lib '/path/to/my/dir';
use DBI;
use Data::Dumper;
print Dumper( \@INC, \%INC );
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That will tell you exactly what is in your @INC, and exactly where Perl is finding all its libraries.