My question is this: how do I write a multi-screen mod_perl program?

The same way you should have been writing it from the start ;D proper scoping and careful usage of globals ;)

See Practical mod_perl: 2.6.1. Porting Existing CGI Scripts to mod_perl

See also CGI to mod_perl Porting. mod_perl Coding guidelines

perhaps I am misunderstanding the use of handlers, or the configuration of mod_perl. What I would like to avoid is an entry in httpd.conf for every page (and this is where I think I might be misunderstanding something.).

Yes. Before mod_perl, I might start a program with a nice template (commentary on this template) like so:

#!/usr/bin/perl --

use strict;
use warnings;
use CGI;

Main(@ARGV);
exit(0);

sub Main {
    ... # CGI program here
}
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After mod_perl all I need for a handler is to rename MyProgram.pl to MyProgram.pm, and rename Main to handler

package MyProgram;
sub handler {
    CGI::initialize_globals(); # unneeded if using CGI->new
    ... # CGI program here
}
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Now, I might also use CGI::Application or Catalyst, so sub handler would consist of

MyApp->new->run;
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or

Catalyst::ScriptRunner->run('MyApp', 'Apache');
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