Your engine is 128T02-3256-B1.
It looks like you have a primer bulb on this engine. I'm sure you are, but if you aren't pressing that, your problem is there.
I personally do not think it is a needle and seat issue, because you said that it runs fine until shut off. In my experience, an engine with a swollen needle and seat will run for just a short time before dying. Because it is running fine once on, I suspect a primer issue.
If you have a primer bulb do this:
Take off the air cleaner (one 8mm bolt at the top of the airbox, or use flathead)
Press the primer bulb a lot of times while looking down the throat of the carburetor. If you see gas start to accumulate in the carburetor the primer is working correctly. If you do not, the primer isn't working correctly. Of course if this is working and gas pools in the carburetor wipe away any excess gas for safety reasons.
I bet you'll find this isn't working.