I would look at the gasket on the carb bowl. If it is dried out it may not seal, and let air in and screw up how the engine runs.
Parts Tree on the internet sell carb parts.
I took one Tecumseh carb bowl off and found parts to a paper washer that was installed on the bottom of the carb between the jet and the bowl nut. Darn thing just fell apart. After cleaning, put it together without it, and it runs fine, and does not leak.
Also there is a small hole for the idle air just behind the choke plate, if that is clogged it may act up.
The idle screw is located on the side of the carb. You want to remove it while you clean out the idle passage. There are two type of idle screws. The ones WITH the black plastic over them are NOT adjustable, and are installed turned all the way in. The ones without the black plastic cap are adjustable.
You probably should take the carb bowl off and clean it out.
If you had the gallon carb cleaner setup with a basket in a gallon of carb cleaner you could let all the carb parts soak for a couple of days. And then clean it and assemble.
There are some videos on YouTube about rebuilding Tecumseh carbs.
Pay attention to the direction that the hinge of the fine wire little spring for the needle valve is pointing when you take it apart. Some manuals for Tecumseh show that hinge in a certain direction.
Some Tecumseh carbs do not have the plastic seat for the needle valve. If yours is like that just leave it that way.
Purolator inline fuel filter no. F21124 works well as an inline filter to keep small stuff out of the carb. You can get them at NAPA and Advance Auto Parts. If you get one try to get a box that still has the two small hose clamps with it, and install it so the incoming fuel flows into the part of the filter that you can see, so you can see how much dirt is in the filter in the future.
In the future a little blast of WD-40 down the idle air intake hole before storage is a good idea. Some people also temporally remove the carb bowl to get all the fuel out before storage. It will not hurt anything to hit the carb parts with WD-40 when you do that. Kind like the old chicken soup cure for a cold. Won't hurt, and may help.
I have been using RedLine SL-1 fuel treatment for keeping fuel systems clean, and Sta-Bil for fuel storage. You can get RedLine SL-1 at Pep-Boys, and you can get a 100 ml graduated cylinder to measure about 4.45 ml per gallon from McMasterCarr either online or by phone.