Most of it works fine, though I had to comment out instances of CGI::use_named_parameters. (Will that break anything??)

"Version 2.57 ... 6. Removed use_named_parameters() because of dependency problems and general lameness. "
You need to know what use_named_params does before you can know if it will break something.

There is at least one section of the site that does not work. When a form is submitted, a query is run. The script calls itself, and then draws the results page (I think) based on the presence of certain variables. The results page draws the template areas of the page, but nothing in the content section, which *should* contain some hard coded HTML, and the results of the query. No errors are reported.

Is there a test-suite? If there isn't one, write a test suite with the old perl, make sure it works, and then try it out on the new perl, and see what breaks.

First, I'm pretty new to perl. I've done some simple stuff, like search and replace, and minor text formatting, but nothing even remotely advanced.

Then maybe you are not the right person for the job?