Here are a couple more items to check while rebuilding the carb. I will edit the first post with this information when I'm done.
First, before you do anything check to see if the throttle shaft is worn out. The shaft will move a lot from top to bottom but the side to side movement is key. Simply move the shaft side to side and see how much movement there is. It will move a small amount but if it can rattle around in the bore then it will need bushing installed or replace the carb. The Kohler service manual has very detailed instruction on how to drill out and install bushings. A worn out throttle shaft will make the engine run lean and make it very difficult for it to run smooth at idle.
If all is good with the throttle shaft, I like to leave the choke and throttle shafts in the carb body. It reduces the risk of having one of the small screws that hold the butterflies in from vibrating out while its running. These tiny screws can cause damage to the engine if they get sucked into the combustion chamber.
On the end of the carb that bolts to the engine, you will find a very small opening in the bore of the carb. This is part of the idle circuit.
This opening will get plugged with sand. I like to take a small gauge multi-strand wire and insert it into the opening to make sure it not obstructed.
Here you can see the idle screw and small opening are directly in line with other.