Your engine is a 19 cubic inch (311 cc) fourth design generation side valve model with vertical crankshaft, flo-jet carburetor, plain main bearings and a 12 Volt gear drive starter with alternator.
My first guess as to why it suddenly started emitting white smoke, is that you over-filled it with oil. You said that after running it for a while, the oil level is still above the full mark, and the white smoke has decreased. It is likely to continue to smoke until the oil level gets down to where it should be. Remember, you can only measure oil level with the engine on perfectly level ground, or the measurement means nothing.
The best and fastest way to sort out this problem is to get the engine level and see whether it has too much oil in it. If it has, drain just enough oil to get it precisely to the level mark. Overfilling is not really good for your engine, since it causes oil to enter the crankcase breather and be ejected into the engine at the carburetor. The engine will probably burn it without much trouble if the level is only slightly high, but meanwhile the crankcase is full of oil froth and this aerated oil is not ideal for lubricating the engine.
