On FreeBSD, /usr/sbin/sendmail is a symbolic link to /usr/sbin/mailwrapper; it doesn't contain the Sendmail binary. How should I replace Sendmail with Exim on FreeBSD?

There is a file called /etc/mail/mailer.conf which selects what to run for various MTA calls. Instead of changing /usr/sbin/sendmail, you should edit this file instead, to read something like this:

sendmail          /usr/exim/bin/exim
send-mail         /usr/exim/bin/exim
mailq             /usr/exim/bin/exim -bp
newaliases        /usr/bin/true

You probably also need to edit /etc/periodic.conf; see Q9202