I would suspect some water in the gas.
Have you thoroughly drained the fuel tank & carb bowl?

Hopefully the tank and lines were cleaned, as Bill said, your problems can lie there.Fuel should also be new. HM 80s have a bit of an issue at low speeds with the sudden chug chug black smoke as fuel can puddle in their strange intake manifold. Yes, that can also be a float issue- such as a small amount of gas in float or sticking where it pivots on the pin. Possibly the float is set a bit off.
If the governor is trying to open the throttle, or you can manually do this when under load, that will confirm if you are dealing w/ a carb that can't keep up. It is a bit sketchy in my head, but some of these had a removable fuel pipe above the main jet that certain styles had to be replaced (per tecumseh) with a service one. Did a Tecumseh dealer do the rebuild? Use a full kit?
Usually the shafts stick due to the carb going green from fuel deteriorating. If there was corrosion in the carb, that doesn't go away w/ carb cleaner and better money would have been to replace the carb.

Fuel tank was dry when got the machine and I replaced the fuel line. The shop that did the rebuild told me that they did a complete rebuild. I ran Sea Foam in the first tank. I'll start with check the float operation and setting, hopefully I'll find something there.

You should check the quality of the spark. I had a snowblower with a 5hp Tecumseh that had similar symptoms as yours. It would start right up and run fine, but sputter some during idle and stall when engaging the auger at full throttle. The spark was consistent but weak. Incidentally I'm working on a HM100 with weak spark.

Anyway, a spark test is quick, easy and narrows down the problem. If you don't have a tester, they cost less than $10 at any big box auto parts store.