The fuel solenoid has nothing to do with your flooding problem. It only blocks the Main Jet to prevent after fire on shut down.

Your problem is with the float needle leaking through.

Sounds like your sump is mostly full of gasoline instead of oil and it's simply getting past the piston rings.

You need to FIX the carb and change the oil & filter.

You could install a fuel shut off valve between the tank & carb.

Thanks Bill. I bought a complete carb overhaul kit and installed/replaced old needle valve and all gaskets. I also got a new oil filter and air cleaners as well as new spark plug. The oil didn't appear to have any gasoline in it but it needed changing. I power washed the muffler, as it was dripping of oil.

The engine still won't run at a constant speed and when it does it runs fast for a few seconds then powers down then revs up and keeps completing this cycle.

I am missing something but not a clue what. Haven't heard much good about B&S engine on web sites but that's just internet talk.

Any suggestions? Thanking you very much in advance (Bill and others that may answer)

The carb probably needs a thorough cleaning. Likely some internal passage(s) are clogged.

ditto bill's advice. Get a magnifying glass, a piece of small dia stiff brass wire - a half-gallon of liquid carb cleaner and fresh rebuild kit. Fully disasm carb, let soak overnite in carb cleaner. Using magnifier and wire, probe every orifice in each component, blowing each clean with compressed air. Rebuild carb using fresh kit components (following included instructions).

I completely disassembled the carb and soaked it overnight in a solvent [a diesel engine repair place down the street let me use theirs].
I dried all the parts off and used small piece of wire to go through all holes and then re-assembled it.
Now won't start unless I put my hand over the air-intake or use a starting fluid but as soon as the starting fluid is used up the engine stops. I put the [?] choke butterfly in so it is open unless I put the throttle level all the way forward and then a wire like big paperclip closes it.
I adjusted the, I think needle valve for idle, from lightly seated to 1 1/2 turns out. Still can't get it to start on its own.
This makes two carb repair/overhaul re[pair kits [both from a Sears Warehouse].
ItâÂÂs apparent it is something I am doing wrong but I donâÂÂt know what.

Make sure the pin on the fuel solenoid is retracting when power is applied.
Make sure the Main Jet is clear.
Part 117 on the Briggs IPL.

Thank You ALL for the advise. I redid [everything] for the third time following everyone's, very helpful knowledge, and finally it started and runs. Has a bit of smoke in the exhaust but I figure its a rich carb setting. I will deal with the smoke, at least it is running smooth and consistently.

Again, Thank You ALL.