Is there a way that I can tell the Email::Date::Format module to use the version of Exporter in my lib folder?

Yes, by telling perl from your script

use lib '/absolute/path/to/my/lib';
use Email::Date::Format;
[download]

I've event tried editing Format.pm and changing

In that case try replacing  use Exporter 5.57 'import'; only with  use Exporter ; and nothing else

use lib...; only belongs in scripts

OTOH, 5.6.1, the perl you seem to have, is 12 decades old (120 years), you can install a newer perl like citrusperl in your home directory, or use perlbrew or perlall to do the same from sources