Briggs Runs Too Rich With AirFilter

I have a craftsman edger with a Briggs 500 series engine. Model #10t502.
It has the plastic pulsa prime carbs on it. The engine fires without problem, runs fine for a few minutes and then starts to act up. It sounds like it’s missing and you can smell it’s running rich. It will do this intermittently... runs fine for 10 seconds, sputters, runs fine for 15 seconds, sputters... and so on.
Here’s i what I have done:
Carb clean with diaphragm and gasket replacement
New spark plug( previous plug had carbon built up)
New airfilter(oiled foam)
Gapped the coil... the previous owner actually had the coil touching the flywheel and I fixed this when I regapped the coil. There is some scoring on the fly wheel from this.

With all that nothing has improved. Because I could smell it running rich I took the air cleaner off and the sputter went away and the engine runs steady, it also doesn’t have that rich smell. As soon as I put the air cleaner on sputtering comes back. These are oiled foam elements and I thought maybe it was over oiled so I put a brand new non oiled filter in and tried running it with the same sputtering as an outcome. I then oiled the filter as usual.

I also pulled the kill wire off and that didn’t help. The only thing that makes it run correct is to take the air cleaner off.

Not really sure where to go from here. Maybe the coil was damaged from when it was rubbing the flywheel but why would taking the air filter off make the engine run correctly.

The carb is stuffed.
Lots of shaved monkys posted You=tube videos on drilling out the jets on these carbs to improve their running.
So either the DPO has acted on one of these or the main jet is missing.

Main jet is there, I should say the machine is only two years old. I have never had good luck with these tank mounted carbs, was hoping to not need to replace the carb but willing if needed

Dumb question, you did mount the new diaphragm against the tank, then gasket, then carb? Those carbs are too simple to run rich, most of the time they run lean.

Yep got the order correct

Try it with the air cleaner screw back in the carb. I'd guess the carb or tank surface is some warped and the diaphragm isn't seated fully.

As noted earlier, the diaphragm only pumps fuel into the bowl.
The bowl has an oveflow so can not flood the carb
They usually run lean unless some one has opened out the main jet.
Seen lots of videos where the idot on the end of the camera insists they know better than Briggs and you have to drill out the jet.

I haven’t had a chance to just put the screw in without the filter however when messing with it in the past I noticed that the tighter I turned that screw in with the filter on , the worse the engine ran. Make any sense? How would a warped carb cause a rich condition?

Install the air cleaner assembly on the carb with no filter in it, just snug enough to hold it in place. If it runs better I suspect you have over oiled the filter. When oiling this filter you must squeeze the filter as hard as possible a few times, until no oil comes out.

FWIW
I install them mostly dry.
I dip my little pinky in engine oil then wipe it across the top of the foam so only the top section is oiled & the bottom is dry.
Good thing is I can get a snorkel for most of the applications that use the oiled foam so the engine draws cleaner air with less dust.