Every Word Is Three Letters

This reminder puzzle came with the nonsensical-seeming line of text "Actually, every word is 'drawn' or 'tornado'." At this point, every single letter swaps things in my understanding of what's going on.

• 28-30 = Latin name for the order with cats and dogs (9)
• 19-21 = Gently trying to influence (7)
• 16-18 = Me, of this puzzle (7)
• 1-3 = Insect or game (7)
• 43-45 = One who rides a cycle (7)
• 34-36 = Person charged with a crime (9)
• 13-15 = Number of US states (5)
• 40-42 = Bee homes (5)
• 31-33 = The quality of being impossible to defend (15)
• 37-39 = To make with machinery, or to make up (11)
• 10-12 = Type of linen cloth, from Latin for "hand towel" (11)
• 7-9 = To come up with a new name or logo for (7)
• 22-24 = Element with second-highest boiling point, after tungsten (7)
• 4-6 = To move stealthily, or to stay out of sight (5)
• 25-27 = Day before today (9)

Key: (12) = unused, (13) = a, (14) = b, (15) = c, ..., (58) = x, (67) = y, (68) = z, (78) = unused

3 5 2 4 4 2 9 9 (4 3)