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They should stay way over the danger zone if tightly wrapped in foil, then a couple of towels and in an ice chest. Actually you will notice how much better they taste with that nice long rest you are going to give them. Also they are fully cooked so unless you use aluminum foil you had the raw chickens wrapped in and towels left over from your son's gym class you should be pretty safe.
Yes you can. Keep your meat probe in it so you can monitor the temp. If it gets below 135 and your more than an hour or so from pulling it. You would need to take it out & pull it, or put it in a 175 degree oven to keep the temp above 135.
I have kept food in coolers for 12 hours safely. Here is the key - put the towels in the microwave and heat them up a little then make sure you fill all the empty space in the cooler with towels or newspaper so there are no voids and believe it or not they will still be hot to the touch when you pull them out