I have a '59 Nash Met. with a straight 4. It runs great, but when I turn it off the engine continues to run for up to a minute, sometimes violently shaking as it comes to a halt.
Any idea what causes this or how to fix it?
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I would check the idle speed first. Many older cars had a different idle speed for manual or automatic. Look for a vacuum leak as this could cause a slight idle speed increase. As could dirt holding the throttle plates open. Check the timing at the correct idle speed. If the timing is advanced the pistons can get hot spots. If the idle is high enough, after killing the ignition the engine spins enough to draw in fuel and the hot spots ignite it, keeping the motor running. This is referred to as dieseling (combustion without an ignition spark) or engine run on.
It sounds like it's dieseling to a stop. As @mikes suggests, there are hot spots in the combustion chamber. This is usually because there is carbon buildup there. You'll need to clean the carbon out, plus check your timing, and also ensure your carb isn't dumping unwanted fuel.
To clean the combustion chambers, I'd suggest a Seafoam treatment. Also, you can try cleaning your plugs (or outright replacing them). (EDIT NOTE: I see in your other question you already replaced the plugs.)
Considering it has sat for a while, rebuilding the carb would probably be in order. If it is an original style carb, it shouldn't be too hard to rebuild. They were pretty simple and straight forward back then. I found a parts catalog online, which may be of assistance. I see they have a parts kit for an original Met listed.