If the backplate is leaking or the carb is leaking you will never get it tuned and it will overheat.
Just so you know you are not experiencing an air leak, get some Permatex copper silicone sealant. Pull the backplate and clean with some carb cleaner. Put a thin coat of the copper sealant on the crankcase part of the motor where the backplate touches it. When you are done just run your finger around the inside to get the copper silicone off the beveled edge, that will insure no sealant in the motor. Put the backplate back on and tighten real good using criss cross pattern on the bolts to ensure an even tightness all the way around.
Now pull the carb out and clean that as well. You want to put a thin bead of silicone sealant on the carb itself right on the o-ring at the base of the carb (last part to touch the crankcase when you put it back in). Make sure you are pushing the carb in when you tighten the nut that holds it in place and make sure it is positioned correctly.
It is best if you let that sit for a day to make sure it all cures. Some grease or petroleum jelly on the needle o-rings and you should be all set there.
Now you have to make sure the needles are set to factory. Hsn is 4-4.5 (I think) and the LSN is 7.5-8. If you have the lsn set at the 10.5-11 like the manual says you will never get it to run right. I doubt you are there because if you were there is no way you would get it to run period.
Try that and an mc-59 plug and you should be dealing with a whole new critter. If that doesn't work, the only thing I know to tell you is to ship it to me, I will see what I can do and ship it back. Like I said, I love my picco and it holds a tune without any problem. I did all the above before ever firing it up. Good idea with any new motor to seal it up and check for any surprises like chewed up o-rings and such.
Keep us posted. That copper sealant should be available at any auto parts store.
Glenn