Take a pic inside your intake looks like you have as much dirt inside the carb and
Engine as you do on the outside, where is the air box/filter?
Whats with the cable I bet you will find dirt on top of the carb slide.
Whats with the boot by the carb looks like a defect by the red arrow.
Looking at the dirt stuck to the carb body your a candidate for a prefilter setup on that machine.
If that was my machine I would seal up the intake really good then wash the Engine good then pull the carb and reed cage look for dirt ingestion before running the Engine.
I would also go through the carb and make sure the entire fuel system is clean, no water or dirt in any part of the system.
Cant beat the price you paid for the 350, you need to lose the K&N filter setup not sure if you found this one yet when you were lurking on the site.
viewtopic.php?f=1&t=5491Others can show you the proper filter setup for the 350 if you want to put dirt ingestion worries to a rest.