The problem sounds to me like your choke is on. My Puch and all mopeds I have driven, when choke is on have the simptoms you describe. Usualy you pull the throttle all the way (and with a bit of force) and the choke should click off. If this does not work there should be another way to turn it off. If you replaced the piston and rings, focus your attention to the carb and the controls that regulate fuel and air mix, you are running to rich by the sounds of it.
Morini Mo-2 (thats a two speed never driven one but they seam to be good engines. keep at it, you should get atleast 40km.)