Tecumseh engine wtpxs.1851ba. starts and runs 15 seconds then quits. I can keep it running with my thumb over the carb like full choke. I have rebuilt carb but still runs only 15 seconds. What else could be wrong?

Many times with a gas mix, after sitting for a while, the mix separates and gums up the carb. Taking the carb off and soaking it overnight in a parts or carb cleaner and running carb cleaner through the tank and gas lines will cure this.

The Tecumseh snow motors are notorious for idling rough. Especially the last few years of production. Most service techs just say "live with it" But there are a couple of things you can do to help.
It is not a carb adjustment issue. It's usually a fuel, clean carb issue.
First, use a premium grade fuel without ethanol and a fuel stabilizer. This will give you the best fuel mix.
Second, Remove the carburetor shield and clean the inside and outside of the carb with carb cleaner. Even though you can't see the carb usually, they do get varnish build up and cleaning it will help some.
Third, Put the recommended amount of fuel system cleaner in the gas. I use Sea Foam brand. Over a tank full or two you will notice a difference in how the engine idles.
Fourth, Use a high quality spark plug and use a new one every year. Make sure the gap is correct.
Good Luck! Feel free to as any other questions you may have. Don't forget to rate this answer!

If it starts at completely cold to fine tune the high jet adjustment.first off the high speed jet should be initially set at 1 1/2 turns out from lightly bottomed all the way in. next when you do your prestart routine,choke all the way on,primer pumped two times,throttle set at the high speed position,ignition key on,fuel line turned on.(this is my prestartup for my 8hp tecumseh)yours may be slightly different.use whatever starts your machine the easiest! once my machine starts up i can almost immediately turn the choke completely off. i can do that because the high speed jet is opened enough or rich enough for the engine to run.if you adjust the high speed in it will lean the motor out(you found that to be true!) to where it will miss and backfire and eventually quit. start at 1 1/2 turns out and start the engine,once the engine starts if you take the choke off right away use what it does as a guide to high speed adjustment. if the engine quits turn the screw out 1/4 turn and so on.use the very first cold startup as the test for high speed mixture adjustment. at completely cold startup you have the jet adjusted correctly when the engine will sustain itself when you completely take the choke off very shortly after startup. let the amount of choke you need at cold start to keep the engine running be your guide.too lean and the engine will not stay running without the choke at all,too rich and the engine will not pull under extreme loads and will show signs of black soot out the exhaust. you want the jet adjusted so within 5 seconds of cold start the engine will stay running with no choke at all. find that needle setting ( don't ad any more!! ) and you'll be good to go.keep in mind the above is not optimum and is a little rich but you will like the way it starts,stays running,and overall performs as adjusted above.

plugged carb

If you want a 'drop-in' replacement, you will HAVE to source a carb that was fitted as original equipment. Following the demise of Tecumseh's engine division, supply is limited to 'old stock', but there is fair spares aqvailability at present. A newer 'primer start' carb will fit, but you will have to remove all the choke operating mechanism. Don't fit a carb from a larger or smaller engine - the mixture will be wrong, and the engine will never run properly.

When you get it started, move the choke to the middle, then let it warm up. you might have to move the choke back and forth until it warms up, keep the throttle wide open.

Not the solid state ignition. If this were the problem the engine wouldn't start at all. There is some adjustment needed to the carb but without more info, I'm not sure what.

Recommend completely draining the fuel system. Tank, lines, and carb. Make sure everything is clean and dry. Clean or replace fuel filter. Always try to store it with a full tank of fuel. It is common to get condensation when the weather goes from warm to cold and back.

i did and thanks - just picked up one that does the same thing and i plan to take your advise

Thank you Jerr