Its a very precise thing to drill out jets as the orifice must be smooth. You may be able to do it with a wire-size bit set and a pin vise. A dial or digital caliper could come in handy. IIRC, jets are calibrated in 'wire-gauge' already, so the wire-gauge bits are a natural for this job. You can use the shank end of the bits to determine the existing size and simply go up one size to start with. Try that and see how it runs. You should actually mow with it for at least ten minutes to get a good assessment of the results. Obviously you have to remove the jets to do this so you don't get any debris in the carb. Very carefully remove any burrs on both ends of the orifice before re-installing. The idle and hi-speed RPM may increase when the mixture is enriched. You adjust this with the knurled disc at the base of the air vane on top of the carb. This is the governor spring pre-set and it moves both lo-speed and hi-speed limits at the same time, so you will increase or decrease both ends of the speed range at the same time. Four clicks on this adjustment (clockwise for faster) is pretty much the max.

The float is what moves the needle and seat open or closed. If the float doesnt lift up with the fuel level it will never cut off fuel flow, thus leading to flooding or fuel leaking when shut off. The fuel valve can circumvent the leaking problem but wont help the flooding issue.

Find the part number and check your nearest Lawn Boy dealer or e-bay. The early version of this carburetor had a hex adjustment knob for the governor spring.If this is the one you have the part number is 609630 for the air vane/throttle shaft and the throttle disc is 609632. Make sure you find out where the old disc went. Here is a link that might be useful: Lawn Boy

Okay, let's see; 89 octane fuel is okay. We need to know exactly what oil you put in and how many ounces. Seafoam is okay and 1 oz/gal is not excessive. A 2-cycle will smoke from the exhaust if there's too much oil in the gas. The 10550 has the 'Duraforce' engine that requires a 32:1 mix of Lawn-Boy oil. That's one 8 oz can to two gallons of fuel. Re-reading your post it sounds like you picked up two small bottles of 2-cycle oil, probably a synthetic or synthetic blend, possibly designed for a 50:1 mix. This actually should have been okay IF the mix was properly agitated before putting in the mower. The procedure on these hard-to-mix oils is to add the first bottle to about half a gallon of gas and try to mix it the best you can without slinging gas around. Then add about another half gallon, pour in the second bottle and mix again. Finish filling the can, cap it and shake vigorously. Shake the can before fueling the mower, every time. If I had to guess, I'd say you possibly did not have the oil well-mixed, and the gas you put in the mower was too light on oil. This caused the engine to overheat, usually causing permanent damage. At least it didn't seize, which is a hopeful sign. All Lawn-Boy two-cycles have iron cylinder sleeves, so perhaps it's not too bad. Check the air filter - remove it, wash it, dry it and oil it with 2-cycle oil or special filter oil. I'm trying to figure how smoke could come out of the carb. If you did scratch up the piston and bore, the compression would be low, possibly explaining the poor performance after the incident. This one thing probably explains the constantly-dwindling sales of the 2-cycle mowers leading up to Toro finally pulling the plug - all it takes is one time 'straight-gassing' a 2-cycle and it's junk. If they had been a hot seller, they would have found a way to make them compliant with emmissions laws. As it was, they decided to just get out of that game since the LB share of the market was so small. People today will not pay attention to details like this - they expect everything they use to be a 'refrigerator' - you plug it in and forget it. 4-cycle mowers are easy to maintain and have similar maintenance requirements as cars vis a vis oil changes and air filters.