White Smoke coming out of Briggs and Stratton engine
My guesses without more information would include the crankcase being overfilled with oil, a blown head gasket, or damage to the piston, rings or cylinders.

I doubt that you have the generator running while it sits at an angle, which could let oil slop over into the carburetor float bowl. Although that's a possibility, too, if you have it sitting on a slope.

Since the engine apparently always produced a little while smoke, and since the smoke doesn't start for a while, then doesn't stop for a while after shutting the engine off, it's probably heat-related in one way or another. As the engine heats up, it expands. If there's a damaged head gasket or something like a hairline crack in a cylinder wall, it could open up just enough to let oil seep through once the engine is hot. Or if you have something like damaged or ill-fitting piston rings, oil might start squeezing past the rings once the engine heats up and the cylinder expands.

You mentioned that if you hold your hand in the white exhaust, little oil droplets condense on your hand. The same thing is happening inside the exhaust manifold/pipe, and the heat will cause that condensate to keep smoking until it is all burned off or the temperature drops low enough for it to stop smoking. If the exhaust valve is open at all after the engine shuts off, any oil in the cylinder will burn off and push white smoke out the exhaust, as well. Especially if it's some kind of hairline crack letting oil into the cylinder, which then closes back up as the engine cools.

Of those possibilities, aside from overfill or running the engine on a slope, the head gasket would be the easiest thing for you to fix yourself. You would need a torque wrench and specifications on the correct bolt tightening pattern and torque for the head bolts for your specific engine.

Or it could be something completely different.

Here's a potentially handy page from Briggs and Stratton, but it might be time to call a local small engine repair shop.

https://www.briggsandstratton.com/us/en/support/faqs/engine-emits-white-blue-smoke