There is a phrase “Don’t let the beddy bugs bite”. I know when it comes from. Doesn’t it sound old-school? What semantic connotation does it have?

Question

Answer

Good night is formal (and dry) (and dismissive)
Don’t let the bed bugs bite
Nite nite (night night)
Sleep tight
This is informal (and warm) (and invites a rejoinder)
Semantic overtones? 
Innuendoes?
Possibly. Depends on the situation and the individuals

Answer

good night, sleep tight, don't let the bedbugs bite