165 to 175 is not hot for the 66/68RFE. I am not sure about the burning smell. Could be multiple things from oil blow by to spilled oil/fluid from a recent oil change.
As far as it not shifting. The previous oneer might have driven it like a grandpa. These transmissions are adaptive transmissions and they shift according to how they are driven in the past 100 or so miles. The more aggressive you drive, the later the shifts with more firm shifting and so on. If you have a way to make the trans go into relearn mode to learn your driving then it might adapt to your driving sooner. I would take it easy for the first 30 miles after a relearn and try to go through all the gears as much as possible. That should be enough time for the valve body to know how mutch fluid to add to each clutch pack.