Most
PDA falls into the barfy category, as well as emotional
love letters, touching sentiments, etc.
How to use in a running sentence: "Look at that
cute couple holding hands, kissing each other as the sun sets on the beach. Barf."
But actually, you are just jealous and would not actually mind being in said situation yourself.
Note that "barf" and "barfy" can be interchanged depending on your personal grammatical preferences.