I've got a HM100 on a gen engine. I kept a eye and ear open for the old fully adjustable jet carbs and finally stumbled across one.
The OEM non adjustable carb would cause the engine to hunt and surge when the engine was warming up and worse during temp's below 30 degrees F and if the engine went to running low on fuel it would really hunt and surge. With the adjustable jet carb when the tank run low on fuel the engine just died instead of doing the long period of hunting and surging and voltage going up and down. Not a good thing when loads are connected.
Just a hint about en riching the non-adjustable carb so as to not need a run with choke. ((if all else in the carb is ok)
What I done:
After I seen that the old fully adjustable carb worked great I went to work on the Al Gore carb and enrichened the removable idle jet little bit at time using the tiny micro bits from fleece bay and I also had to get the micro chucks for the bits. I had to do the drilling jet enlargement very carefully by hand so as to not twist off a bit. Them non-adjustable carbs are leaned out and they also use the idle jet area during intermittent and high speeds.
Anyway the enrichment of the OEM carb did the trick. Mine was like yours also, when cold it ran better with some choke and also during cold weather. I did check before I started enlarging the jet that a OEM replacement jet was available in case I needed to go back.
It was an easy mod once I started because the jet could be removed with the carb in place re-sized with drill bit by hand and then re-installed and test. It's the screwdriver slot jet under the little plastic cover. I throwed the plastic cover away. I heard a rumor that the color of the plastic cap was a hint as to the OEM size of the jet if it needed replaced. I think maybe the same carb was used on the 8hp but non-adjustable were jetted different but not sure, but the adjustable jet type carbs were same for the 8-10.
I now have two good Carbs for the HM100.
I also converted this hard to manually crank in cold weather engine to electric start and added the 3amp stator. In cold weather it takes a good at least 275CCA battery to crank good. Larger battery is better.
Tecumseh really got some bad raps for the issues you and I have experienced with them hunting and surging, especially the later models with the lean carbs.