Furnace kicks off at 65/66 and won't kick back on unless you turn it all the way down and back up or turn it off and back on

How old is the thermostat, and what kind is it? Have you tried resetting the temperature you are trying to maintain? Is it set at 66?

Do you know the brand and model of the stat? Before trying anything on your own regarding the furnace I would get a cheap replacement stat and test it to see if your furnace will stay on. You can always return the replacement stat if its not needed