Only thing I can think of is a partly worn metering diaphragm. If it has lost some of its pliability, it can prop the needle valve open, flooding the engine.
It starts off minor, and gets worse. They will generally run rich with this issues, hence slower at full throttle and less powerful
If this is happening, and there is only minor flooding, fuel mix could be running out the carby throat whilst its sitting in the shed, and if the air filter is tilted slightly down, the mix will run onto it.
The fuel evaporates, but the oil doesn't.
That said, a blocked fuel cap vent could exhibit similar thing - if the pressure cant be released, overcomes the needle valve, and 'floods' the carby. Less likely though.
I also guess that you mean paper filters, not oiled foam filters. Some people have a habit of dunking the foam filter in a pot of oil, and installing it saturated.