Panels everywhere messes up rendering of admin themes

I'm converting a D5 theme based on blocks and regions, to a D6 theme based on panels. I installed this module and configured my default site template to use a 3-column layout. The header and footer are being rendered in page.tpl, and everything looks fine on the front-end. However, my admin theme (currently set to Garland) is messed up, as seen in the attached picture.

I believe the problem might be in my interpretation of how to set up my default template. Should I be putting 100% of the site elements -- headers, footers, menus, etc. -- into the template? I was just using it for the main section between the header and footer.

I can't enable "Override Page Template" because too much of my theming is based on tpl files and template.php preprocess functions. I tried switching the admin theme to other themes and only the simple ones, like Marvin, rendered properly. The others, like Rootcandy and Bluemarine, were either greatly messed up or impossible to use.

Is this an issue with how I set up my template? Should I just switch to Marvin for the admin theme and call it a day?