Will start but when trigger is pressed stalls and stops
I have had that kind of problem myself many times. Its just one of those things that 2-stroke engines just seem to do at times. I always do the easiest possible problems first and then progress to the more involved and expensive problems next.
First: I'd recommend replacing the Fuel Filter, and the Air Filter. as Fuel Filter gets partially clogged it restricts the flow of gas to the carburetor, so when the trigger is pulled, it immediately is starved for gas and stops. When you replace the Fuel Filter always examine the fuel line for cracks or deterioration. Also, it is always a good idea to replace the Air Filter too. If after replacing the Fuel Filter, and the fuel line, if it needs replacement too, try starting the engine. If the problem persists, we'll move on to the next possible problem.
Second: The carburetor has two adjustable jets, one is typically marked High, and the other is Low. You will need a special tool that looks like a screw-driver that has a splined female end on the end, that will allow you to adjust the carburetor jets. If you don't have one of those tools it pays to get one... Amazon typically sells them for Poulon products (which is commonly used by Sears). Once you have the tool, slide it over each jet, with the engine off, and then CLOSE each jet until it is lightly snugged down, and keep track of where the tool is (I mark mine with a marker line on the tool's shaft), and slowly open each jet 2 turns. (with a 27 cc engine somewhere between 1-1/2 turns and two turns is a good starting point. I typically start with 2 turns to make sure there is good gas flow, and the engine will tend to run rich, but it should run.)
Once this is done, try starting the engine again.... Make sure it is idling well, and then pull the trigger, if it doesn't quit on you, you've solved the problem. The most you might need to do is slightly turn each jet in about an 1/8 turn to fine tune the engine mixture. If you close the jets too much, it won't run well again, and will starve for fuel again. Keep the jets balanced and it will serve you well.
Third: If that doesn't work either... Its time to get into the carburetor itself. I've tried rebuilding these carbs, and they are a pain, and by the time you purchase all the necessary ingredients, it costs more than the trouble is worth. I typically find an identical replacement carburetor from RepairClinic.com, Amazon, or even this site, etc. and just buy an exact replacement carb. Its fast, easy to replace, and will solve a host of problems, and you won't have to experience spending a day looking for one of those tiny rebuild-kit springs that flew across the garage floor. Don't ask me how I know this.
I hope this info helps you. Good luck....
Rayko