The danger zone is 40° to 140°, with the most dangerous temps being in the 60° to 80° range. Serve Safe says you have 4 hours before the temp needs to be 40° or below. Your situation sounds iffy to me.
One thing to remember is that house hold refrigerators are designed to keep things cold, not cool them down, hence your warm fridge. I always use an ice and water bath with an ice bag on top to cool the meat down to cold before fridging.
Hopefully others will chime in with more info.