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I have a kohler engine on a walk behind mower you can start it and it will run, but it seems to be running rich because of the self relieving choke on the Walbro LMK carb. When you try to increase throttle it stutters and you can see the exhaust darken like it’s rich. I took the air filter off and when I hold the choke blade fully open it runs fine. But it’s spring closes it back some when you let off.

Ye you are quite right
The auto choke is not turning off .
You need to replace it

That part is obsolete and I can’t find one. Any reason not to just remove the spring and run it without one?

Better still.
Get a universal choke cable and toss that auto choke where it belongs
In the metal recycling bin .

It not a cable. It’s built into the carb.