So I bought a gx340 engine that has been sitting for a year or two...I knew it was going to need the carb cleaned, as the fuel was brown and smelled horrible. In the process of going through the carb I thought of this thread and it occurred to me that I never specifically asked if you pulled the pilot jet out (I think this series engine has similar to the GX carbs) It is a little unusual in that it is external, and plastic. You take the black plastic idle speed screw out, then the black plastic pilot jet can be pulled up out of the carb body. When reinstalling it, the flat side goes against the carb body. Partially plugged pilot jets will 100% cause this kind of surging...
I added an edit to the above last night, but apparently it didn't take. Edit, it looks like the GC series uses an external brass pilot jet, so its a little more obvious than the plastic one on the GX carb... its in the same place though, right under the idle speed screw.