Why do my rooms heat to different temperatures?
I think two things will resolve 95% of your problem. Get a new thermostat (I like my Nest), and adjust your vent dampers seasonally and to suit your space usage. 

Your thermostat apparently has issues with temperature, and a modern "smart" thermostat offers many benefits. One is the ability to run a fan outside of the normal heating/cooling cycles to equalize temperature in your rooms. 

The variance between rooms isn't particularly concerning. Solar gain, airflow differences, human activity, and other factors can explain that. The highest temps are seen where cooking and bathing tend to introduce heat. Tweak your vents and see where that gets you. Be sure to look at the returns, too. They're critical to proper air movement.

Replacing your furnace won't do much unless you spring for multiple zones or automatic vent controls. That's probably not a practical approach for cost reasons.