It would seem somewhat simpler to not even bother with HTTP authorization and just use cookies and my own webapp code to decide whether users are logged in or out

No, don't go there, it does not make sense to have each and every web application manage session or authorization cookies.

The common solution is to setup an 1) Authentication and an 2) Authorization Handler, as Perl modules that are called by the (Apache) webserver upon each url request.

These modules have been written already, an example that you may want to have a look at is AuthCookieDBI.pm

The only thing left with your web application is that you might want to implement a logout button that simply makes a call to the Perl module to have the cookie invalidated. And you will want to create a login page somewhere. That's all.