I would vet carefully info from Thumper Talk. There are some experts there but plenty of clowns too. Stay with the experts.
I owned a DRZ-E and kept a friends S model running for about 10 years, so bit of background on this bike.
With all the well meant suggestions here ... you may end up chasing your tail, headed off to the weeds, opening several cans of worms along the way.
A careful approach is needed now. Think it all through.
I doubt too much oil on the K&N was ever an issue. I hate K&N filters and would never use one ... but it's what you got ... and .. if you keep it clean and LIGHTLY oiled, it will be fine unless riding constantly in severe dust behind trucks or several other bikes. The K&N does not filter dirt well, more oil will not help for long.
The engine may suck a tiny amount of oil in ... but that would soon be burnt up. I'm guessing the rich running/fouled plugs are caused by something else.
I would replace the complete O ring set in the float bowl. I think there are two or 3 O rings, including the one on the float stem where it plugs into the bottom of the float bowl.
Make sure you have not lost any little rubber items.
Most common problems are old O rings (that may look OK to naked eye) and a clogged Pilot jet. The Pilot jet can be hard to clean and may LOOK like it's OK ... If you can, get a NEW one.
A word about Chinese parts. NOT GOOD! Many in the DR650 and DRZ community have had problems with Jet and O ring kits of Ebay from China.
Some may be OK but I know that some are NOT. If you can get OEM Suzuki, I would.
Try these guys to start:
https://procycle.us/bikepages/drz400.html#fuel
I doubt your petcock is the problem. It's more likely the O rings in float bowl.
Make sure you account for ALL of them. You will need good pics of the Carb.
Don't try to work on the Carb with it ON the bike. Take it off.
Obviously bad fuel is a concern ... and has faked out plenty of mechanics including me.
So, really try to make sure the fuel is OK. Water would be the real enemy. Grit and crap can be filtered out ... but water not so easy.
Remember, it's heavier than FUEL so it collects at bottom of fuel tank or float bowl.
Make sure your fuel filters are clean and flowing. I've forgotten if DRZ-S has an internal fuel filter like the DR650 does? The one on the DR650 is hidden and some miss it. Not sure on DRZ400 ... but check on that.
Check the Air Cut off thingy and make sure that little O ring is in there. Procycle has a kit if required.
You may have an inline filter as well, plus your petcock stem is a filter too. All must be checked and clean.
If you carb leaks it will eventually fill your crankcase with gas. Not good! So either the vacuum feed petcock is faulty or the O rings are bad in the float bowl or little rubber "needle valve" thing is worn or has crap collected in it's seat.
Poor quality or ill fitting O rings can also cause problems. The Pro Cycle kits are pretty good but OEM Suzuki is ALWAYS best.
Once you get it running make sure you run filters and filter the fuel you get through nylon stockings and try to find some sort of additive to put in fuel to absorb moisture. Don't use too much! The locals must have something that works.
Sorry for long diatribe!
All the best!