Thanks u/Stock_Requirement564. I looked up the kit with the part number you supplied.

https://www.amazon.com/Briggs-Stratton-801427-Carburetor-Overhaul/dp/B008K522KI

It doesn't look like that kit comes with a seat. Which is probably the main part I need, right?

I found this kit (https://www.briggsstrattonstore.com/801317-Briggs-and-Stratton-Seat-Inlet-OEM/) that has a needle & seat. But this seat doesn't look anything like the seat shown on the YouTube videos I could find about replacing the needle & seat on small carbs like mine. The seats I found on the youtube videos are small rubber disks that are removed with compressed air or a small sharp tool. The seat included in the kit I linked above is brass, and it looks like it screws in, rather than being a press-fit.

I just want to make sure I have the right part. Evidently the seat in my carb is very different from most carbs I came across in YT videos? Also I suspected the problem might be that the seat has swollen due to ethanol in the gas. But if my seat is brass, then that theory doesn't hold. So what might be the problem? Now I'm not sure if I should replace the needle & seat.

Thanks for any help.