Got my first (used) mower with the autochoke, a "feature" that many folks dislike. I, too, had problems with it's operation even though the thermowax part seemed to be working properly. I decided to see if I could simply convert it to manual since I have a bunch of Honda carb parts that I saved from previous jobs. First thing I noticed was that the handlebar lever decal (orange arrows) did not have an indicator beyond the "rabbit" for Choke as on pre-autochoke models.
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However, if you play with the throttle adjustment at the engine (orange arrow), and keep an eye on the butterfly movement through the air box cover opening (yellow arrow), you can cause the black lever to move past the "rabbit" and thus the butterfly will now be closed.
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Next, I replaced the autochoke spacer with a manual choke spacer (below) that I had in my parts stash. It can also be purchase separately or sometimes in a rebuild kit if you don't have one.
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I now have a manual choke which I much prefer.