cosy affair

You can say that, but it sounds like something from the 1950's.

The word "cozy" implies being physically close. A guy and a girl spending the night in a blanket at the top of a ferris wheel would be cozy

co·zy
/ˈkōzē/
adjective
adjective: cosy
1.
giving a feeling of comfort, warmth, and relaxation.
"a cozy cabin tucked away in the trees"
synonyms: snug, comfortable, warm, homelike, homey, homely, welcoming; More
noun
noun: cosy
1.
a soft covering to keep a teapot, boiled egg, etc., hot.
"a photograph of Smith pouring tea from a pot with a knitted cozy"
verbINFORMAL
verb: cosy
1.
give (someone) a feeling of comfort or complacency.
"she cozied him, pretending to find him irresistibly attractive"

Like I said.