If you don't need command substitution and parameter expansion inside your here-document, you can avoid using tabs by adding the leading spaces to the delimiter:

$     cat << '    EOF'
>         indented
>     EOF
        indented
$     cat << '    EOF' | sed -r 's/^ {8}//'
>         unindented
>     EOF
unindented

mail_content="Last Change: $dateLastChanged

    This is an automated warning of stale data for the UNC-G Blackboard Snapshot process."
    mailx -r "Systems and Operations <sysadmin@[redacted].edu>" -s "Warning Stale BB Data" jadavis6@[redacted].edu <<<"$mail_content"

You can change the here-doc operator to <<-. You can then indent both the here-doc and the delimiter with tabs:

#! /bin/bash
cat <<-EOF
    indented
    EOF
echo Done

sed 's/^ *//' >> ~/Desktop/text.txt << EOF
    Load time-out reached and nothing to resume.
    $(date +%T) - Transmission-daemon exiting.
EOF