Your approach goes wrong on several levels.
First of all, you're not supposed to store database handles as persistent variables. It's a bad, bad idea. Use Apache::DBI and mod_perl for that. If you do not want to use DBI directly, I believe DBIx::Class does persistence as well.
Next, it won't work like you expect it to. Say that user 1 comes and accesses a.cgi. Along comes user 2 and accesses a.cgi as well. User 1 and 2 move on to b.cgi, but then user 1 decides to go back to a.cgi. But what user 1 sees is not their data -- it is that of user 2, if even that. This will not go well, and is dangerous for sensitive data.
Instead, what it sounds like you want is a session with a storage backend. Consider CGI::Session, for example, or Apache::Session(X) for mod_perl.
If you want to make something that will go beyond just simple CGI scripts and want something extensible and robust, consider Catalyst or Mojolicious(::Lite). Feeling adventurous? Try Combust, which powers many perlweb sites, IIRC. I personally prefer Catalyst, but you might like something else.
Update: Replaced <pre /> with <code />, because that did not look right at all.
confess( "I offer no guarantees on my code." );