Assumption correct. After you have completed the class and have a stamped and signed (by the instructor) application you fill it all out and do NOT sign it yourself.
Get thee in front of a notary public, show ID, and then sign it in their presence. They will likely also have you sign a register or log-book that they commonly keep for their records. Add a fingerprint card, a check, an envelope, postage, and a few weeks time and you too will have a geeky looking UT CPL ID card with a red stripe on it.
Have fun.
rwc