One example of not so much Getting Crap Past the Radar as the radar being switched off completely was in the pilot episode. When told by Garret and Roland not to kiss a possessed Kylie, Eduardo simply replies "Why not? She's legal!"
In "The Crawler", Janine makes a remark that her relationship with Egon would be easier if love was something he could see in a microscope. We then hear Dr. Spengler say "Interesting. These paramecium are mating!"
In "Casting the Runes", when the team enter Kahlil's dimension, the deity demands to know what happened to his messenger, who was captured by Egon and Janine on the other side. Garrett answers with "You mean Dog Face? He got neutered!"
When Eduardo wishes that he would get the same respect from Kylie as she gives her cat, the wish granting ghost turns him into her cat. When the other Ghostbusters find out that Kylie's cat is actually Eduardo, Garrett groans, "Ugh, I don't even wanna know what he wished for to end up like that."
"The Sphinx" opens with the riddle "What do men do standing up, women do sitting down, and dogs do on three legs?" This is one of those joke-riddles that has an immediately obvious sexual/scatological answer (in this case, "urinate," though "have sex" has been used as a punchline on occasion) and a more obscure "clean" answer ("shake hands," which hadn't been the case for decades when the episode was new).