Okay....let's assume your Sears tiller has a Tecumseh engine (thus a "bowl" style carb, gravity fed fuel). The reason I believe this is your setup is that you stated gasoline would come out that tiny hole on the side of the carb even when the engine is not running. This cannot happen on a Briggs&Stratton engine which has the gas tank under the carb (not gravity feed) and uses engine function to draw fuel into the air flowing thru the carburetor.
This fuel leakage (pouring out as you put it) indicates a sunk (or stuck) float, and no needle valve action (needle not seating). The tiny hole on the right side which you say the fuel comes (pouring) out of is indeed intentionally put there by the factory. That hole is a "vent". It vents the bowl to the atmosphere, that hole has to be there......if the float (in the carb bowl) is stuck down (as I believe yours is) gasoline will flow from the fuel tank to the bowl by gravity, over filling the bowl to the point of fuel going EVERYWHERE. You must also
have a "blocked" (plugged) main (and pilot) circuit in the carburetor because you stated you needed to manually inject fuel in the carburetor (in the air intake?) to get the engine to run at all. Furthermore, if the main, and pilot circuits weren't blocked the engine would be flooded with gas due to the bowl/float/needle scenario I described above. If the main/pilot were fine, your engine wouldn't run no matter what you did because fuel would soak the spark plug/combustion chamber due to the bowl overflowing.
You have multiple problems, and they all sound like carburetor related. This is common with Tecumseh engines (and motorcycles with gravity fed, bowl type carbs) which get put into storage with an appreciable amount of gasoline in the tank. The gas drains slowly into the bowl replacing what evaporates thru that "vent hole" I explained. The residue which accumulates in the carb from years of this neglect clogs things up. The residue is heavy elements of the gasoline which cannot evaporate, and first become a thick, syrup like, stinky gunk. Then harden like dried paint. The inside of your carburetor needs to be serviced by a professional. When they get as clogged as yours sounds to be, I don't recommend the unskilled (inexperienced) fiddle around. I don't recomend you tackle your carb problems yourself. Unplugging the pilot jet, and main jets on small engines which are "this cemented" are particularly difficult. The methods involve the use of pin drills, hot wires, torches, toxic solvents, knowing where and how to drill, poke, and blow to resurrect the carb. I've done dozens of them over the years, and it takes considerable skill. You could buy a new carb for your 6HP Tecumseh on eBay cheaply. There are guys selling brand new carbs like this all the time on eBay. After getting your carb professionally serviced, or replaced with a new carb, BE SURE TO DO A CRANKCASE OIL CHANGE because gasoline probably ran into the crankcase from the stuck float/bowl problem thus corrupting the oil. Use a good quality, straight 30W oil in your 6hp engine during the summer. Do NOT use multi viscosity oil. Good luck.