Re the stutterers and PC issue, I don't see this song as making fun of stutterers. I think it's sort of gently teasing a guy who was so shy and nervous around GIRLS that he would stutter whenever he tried to talk to one, especially one he was "sweet on."   The protagonist (?) in this song is not nececessarily a stutterer in other situations, including the kind of frightening situations he might find himself in as a soldier. He is just very insecure when it comes to approaching the object of his affection.

Flash, that Mel Blanc recording sounds like he was singing it as Porky Pig.
Now, I can understand why Porky and Thylvethter and Ewmew Fudd may offend some people as not PC, because they were characters for whom one of the principal jokes was their speech defects. Put I don't see that as the case with "Jimmy" in this song; the joke is his shyness and nervousness around women he's attracted to.

Rita, I love the "C-c-c-cootie" version!