Based on this part of your description:

It seems to keep variable values from one call of the Perl script to the next one. (the webpage produced by Perl creates a form, which calls back on the script). This script is run on many other servers without this problem.

It seems more likely that the problem has to do with CGI parameters rather than with other variables in the script. You may want to study the the CGI man page -- esp. the section on "Creating Fill-Out Forms", where the 2nd paragraph gives a note about default values for form variables. It may be worthwhile to create a simple test CGI script that can demonstrate the problem, including the server-dependent behavior, so that you can work out an appropriate solution, and then factor it into the larger application.

As sundialsvc4 said above, it makes more sense to figure out how this one server differs from others. You may need to fix your script (rather than fixing the server), but the point is to make sure the fix is based on a proper understanding of the problem.

One more thing, about the concept of "initializing all variables" in a perl script "in one fell swoop". In a well-written script, such a concept makes no sense. At any given moment of run-time (e.g. at the transition from BEGIN to MAIN), many variables in the script are effectively non-existent, because they are lexically scoped to specific, minimal blocks of code that aren't being executed at this moment; you can't initialize something that doesn't exist yet (or doesn't exist anymore).