It would help to know exactly what machine this engine is on.
Right off the bat, I can say there is likely a globule of water slurry hanging around in the bottom of the fuel tank. That stuff collects over years and just slumbers around in the bottom of fuel tanks.
The fuel tank is not easy to inspect or remove on some models, so it is most often avoided until the last possible moment.
Note: a bottle of rubbing alcohol may help, but any trash in the tank will remain behind or get sucked in and keep creating problems.
The next possibility is you could have installed a cheap aftermarket carburetor, and it is running the engine too lean at idle. Or the intake seals/ gaskets are leaking due to reuse.
Getting worse as it runs is a classic symptom of fuel starvation or the wrong spark plug(s) are being used.
I recommend you clean the original carburetor and clear all the small passaged out and blow them clean with carburetor cleaner, then put the original back on if possible.