***" Could the carburetor need cleaned better?"***
Yes, that would be the most likely cause for this, based on your description of how it runs when spoon fed some gasoline.
It is recommended that the carb be disassembled and cleaned with spray can carb cleaner.
Carefully probe all the small holes and passages with a stiff piece of wire (small wire, like wire tags have).
Blow out with compressed air.

Agree with mownie, I cut a piece of wire off my wire brush to clean the holes, Also make sure your float bowl doesn't have a hole in it and it's filling up with gas since you mention it was nasty.

jweber - To add to what mownie said (and you DO need to thoroughly clean the carb), there are a couple other things that could cause your problem.

#1 - Your STX46 has a gravity-feed fuel system; i.e., there is not a fuel pump. If you replaced the fuel filter with ANYTHING other than a Kohler 25-050-21 then the filter might not be letting enough fuel through to feed the engine. If, by chance, you bought the wrong Kohler filter, it would have smaller openings through the filtering media, and, without a fuel pump to force the gas through, the engine will starve for fuel. CHECK THE PART NUMBER ON YOUR NEW FILTER.

#2 - If your tractor is equipped with a fuel shutoff solenoid, it could be malfunctioning. The solenoid (if you have one) is on the bottom of the carb and has a thin electrical wire running to it. If you have such a thing, check to be sure that battery voltage is getting to the solenoid when you turn the key switch on (you should hear a distinct "click.")

Best luck, and post back as you progress through troubleshooting.

I had to clean a carb twice before my F525 would run. Be sure to not rush and do it right the first time.

Good luck!