Plack should act as a middleware between your code and the web server, there shouldn't really be much need for refactoring.
In fact, it looks like it already has a handler: https://metacpan.org/pod/Plack::Handler::Apache2. Sorry, this is going the other way, plack -> mod_perl. Regardless, I think the Plack documentation mentions somewhere about routing everything properly through Plack to your existing mod_perl app code: https://metacpan.org/pod/PSGI::FAQ#I-have-a-CGI-or-mod_perl-application-that-I-want-to-run-on-PSGI-Plack.-What-should-I-do
In terms of persistent database connections, I'm not sure how this is tied to mod_perl. However, I recommend looking at DBIx::Connector as it has a lot of really good connection logic built into it while still being "just DBI".
Three thousand years of beautiful tradition, from Moses to Sandy Koufax, you're god damn right I'm living in the fucking past