That's no good, since $emailaddy is undefined. Fix:

sub manually_page {
    system("/bin/bash -c clear");

    # defines the week number for the current week 1 - 53
    my $wnum = &UnixDate( "today", "%U" );

    my $sched = new Schedule::Oncall;

    open(my $sesame_fh, "sbur.sched.$wnum")
        or die("Unable to open sched file: $!\n");

    my @sched     = $sched->load("file" => "sbur.sched.$wnum");
    my $oncall    = $sched->oncall(time);
    my %info      = $sched->info("$oncall");
    my $person    = $sched->oncall(time);
    my $pemail    = $info{pager}->[0];      # assigned oncall pager em
+ail
    my $fname     = $info{fullname}->[0,1]; # assigns the full name of
+ who's oncall
    my $emailaddy = $info{email}->[0];      # assigns the email of who
+'s oncall

    my %mail = (
        To         => "$fname <$emailaddy>",
        From       => 'oncall script <oncall.pl@dcx.com>',
        Subject    => 'Test message yepper',
        'X-Mailer' => "Mail::Sendmail version $Mail::Sendmail::VERSION
+",
    );

    ...
}
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Why are checking if you can open the file where you do? load should return an error or die.