If there is a way to install a compatible version of DBD::mysql, then I need some guidance on how to do this.

Maybe, but will be tricky. If the new hosting service can't or won't resolve the incompatibility, you should consider a different hosting service

Since it sounds like you have no way to directly run the CPAN tool in your virtual host, you will need to duplicate the Perl installation on your own machine.

Also, you will need to be able to upload not only your Perl programs, but compiled library files as well. And you will need to be able to create the required subdirectories so that Perl can find the files.

Config can be used in a Perl program you run in the virtual host to tell you about the Perl installation.

Using that information, duplicate as best as possible that Perl installation on your own machine. Then you can install DBD::mysql on that. Then you can upload the installed pm and so files to the virtual server.