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.