John,
Welcome to Exim.
You need no changes to FormMail.pl to work with Exim. (Because in this
case, Exim behaves like sendmail. You can just have sendmail binary really
be exim or use the path to exim.)
A great thing about Exim is that it has excellent logging and
debugging. Try to find your exim logs (maybe under /var/log/exim) and look
at a file named mainlog (or exim_mainlog) to see what happened to the
email sent to it.
For your information, various versions of that FormMail.pl have known open
relay issues (they are abused to send spam). So make sure your version is
up-to-date or search for an alternative.
Jeremy C. Reed
http://bsd.reedmedia.net/ -- BSD news and resources
http://www.isp-faq.com/ -- find answers to your questions