Rough idle, backfires, stalling and smoke

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The running well in cold weather and bad when warmer indicates a choke issue. The cable may need to be readjusted to allow the choke flap to open completely when the choke is off.

Tuning the car at idle should also help. Connect a vacuum gauge to a vacuum port on the intake or base of the carburetor. Allow the vehicle to warm up to operating temperature. Locate the idle mixture screws (there are two usually) and adjust them in or out until you get the highest vacuum reading. Repeat this on the other screw.

Another thing to check is your ignition timing. Again with the car at operating temperature you will use a timing light to see where the line on the balancer lines up with the timing marks usually to the right side of the front of the engine. the timing should be 5 degrees before top dead center at 550 RPM. You may have to adjust the idle speed a few times as well as readjust the air/fuel mixture screws. Always check timing with the vacuum advance disconnected and the line plugged.

You mention it rocking which can indicate a misfire. Make sure all spark plugs are clean and not fouled and check your plug wires with an ohm meter. Place one probe in each end. They should read around 1000-1500 ohms per foot of wire.

You may also want to check for vacuum leaks. examine all vacuum lines for cracks or breaks and spray some carburetor cleaner around the intake where it meets the heads and where the carburetor meets the intake. if idle speed changes you have a vacuum leak. I'll post a link below on that.

https://www.2carpros.com/articles/how-to-use-an-engine-vacuum-gauge