Yep, 390323. I bought it on ebay
New Carburetor for Briggs Stratton 390323 394228 7 8 9 HP Engines Carb | eBay .
Thanks for the link. I read the page and the PDF, and so it appears that the problem resides somewhere in table 8, 9, or 11 but could also be in table 4 or 5 (due to sometimes having black exhaust smoke).
I don't think the issue is related to pre-carb fuel supply UNLESS it's the gas cap, however, I don't think it's the cap because it behaves the same with 1) the cap screwed on tightly, 2) the cap completely off, and 3) partially on. It may not be the proper gas cap though -- it doesn't thread on properly...
I don't think it's an air related issue as it behaves the same with or without the air filter assembled (even after dousing the pre-filter in oil and squeezing in clean cloth).
As an additional note, when the main jet is opened about 2 or more turns, the carb appears to leak a very small amount of fuel either from the lower mounting bolt or the screened hole next to it (vent?).
I'm thinking it could be related to the float in the carb, but it seems unlikely as the problem is the same with 2 different carbs.