Sounds very much like crankcase seals, Classic symptoms, impossible to start when left standing for a long while then once started runs very well.
And the flooding most likely caused one of the seals to pop.
So it is time to visit a workshop or do an all day repair for a 1st time DIY.
However you can really only confirm seals by doing a pressure & vacuum test on the engine.
To do this on a Stihl you need some special tools and blanking plates.
Go to You tube.
Search Crankcase pressure testing.
The upload for Joe Pace ( ex-Husqvarna dealer service school ) is the best one.
Before you start, pull the muffler and look at the piston.
If it has deep scratches that go right across the rings then the barrel & piston is also toast.
Blown seals cause lean running.
On a chain saw or brush cutter the operator notices the lack of balls but on an auger with the big reduction gearing it is no where near as obvious.
Well that sucks I guess its off to the shop any guess ballpark on cost?
I charge out scheduled rates ( warranty work ) and seals is 2.0 hours ( at whatever they charge ) + parts.
A small workshop ( without a massive glass fronted showroom ) will generally charge less than a franchise dealer cause they are free to use after market parts.
You know how it goes, Oregon gasket $ 1.50 Stihl gasket $ 4.95 ( same gasket different packaging ).
Pull the muffler off first and check the piston.
Do not tell the workshop you suspect the seals, let them do their diagnosis.
I have just finished doing an 034 that I was told the seals needed replacing so I replaced the seals.
The real problem was a leaking impulse line which caused the crank to fail pressure testing.
So a $ 50 job became a $ 200 job because I trusted the diagnosis of the Stihl Shop who told the owner than it needed new seals and that the seals are NLA.
And the NLA bit is true, Stihl no longer list the seals . The Stihl shop can not fit after market seals because it contraviens their franchise arangement where as I can fit freely available std seals.
On your engine the impulse feed goes through the carb manifold then into the fuel pump in the carb.
It was an illustrative example .
On your engine you a fuel line and a return line.
The fuel supply line has thicker walls and goes on the barbed fitting.
The smooth fitting is the return line and is generally a smaller diameter tubing.
Diagnosing small 2 stroke hand helds over the web is near impossible, in fact it is hard enough when you have them in your hands.
Right now I have a K970 saw that has been in the shop for 5 months which is busting my balls and I have that, on the bench, pressure testers , compression testers , ignition analyser, even popped it on the silly scope to check the wave forms of the spark and I still can not get it to run properly so don't expect any one to work miracles on line.
Your engine is a vertically split one so the seals should just push in from either side.
However without seeing , hearing, feeling & smelling it working out what is wrong is a little difficult.
When the magnets in the flywheel get weak or the insulation on the ignition coil breaks down they can be a bugger to start as well as you don't get enough spark to fire at cranking speeds.
You did not give us much to work with but from what you stated, seals or impulse line would be the most likely.
You could check that the 2 bolts holding the carb in are tight cause they will cause it to run lean as well & be a bugger to start .