When you choke any engine it will burn oil because the rich mix washes oil off cylinder walls. Is it possible your carb is too rich? Take a look at your plugs. If you are burning oil, there should be deposits on plugs instead of a clean gray look.
Had a 70 dodge spirit with bad exhaust valve guides. It was a problem with the 3.0L engine design. It would produce a smoke screen. So as mentioned, it could be a valve guide problem. If it's exhaust guide, the plugs will not show it's burning oil, because the oil only gets into the hot exhaust gasses.
I have a 20 hp kohler engine. Had to go with the next hotter plug to keep carbon off plug. Engine doesn't use a drop of oil in 75 hours. After that it starts using oil. Time to change oil. It too will give off a puff of smoke when I choke it.
Keep looking, you will find the problem.