not really sure what the issue is that your having. anyway sounds like 2 out of 3 650sx's my friends all bought at the same time because thats all they could afford. it sounds like a fuel issue to me which wouldnt surprise me because like i said out of the 3 of my friends that bought a 650, 2 of them took a crap on them as soon as they put it in the water, i would honestly start with a carb rebuild kit. there is a FREAKING PERFECTION to the carbs on thos things. when you pull the carb to replace everything that comes with the carb kit you need to remember THESE EXACT DIRECTIONS. when you replace the float needle and the metal swivle piece that holds it, take the screw that holds the rod to the swivle piece and screw it all the way down. once it is buttoned down, give it a 1/4 turn out. the carb requires a 1-2mm height (and thats about the right height) to actuate the gasket on that sits on the plastic piece that sits on the opposite end of the float needle. cleanthe jets obviously. then when you button everything back up, there is a likelihood that your carb is out of adjustment seeing as how you just bough the ski. i believe the stock adjustments for the the top screw is 5/8 turn out from closed and bottom is 1 1/4 out from closed.
like i said these skis IMO are junk but they are good learner skis. both my buddies skis to run off of thos adjustments. after countless hours of pulling off fuel lines, replacing parts, pulling things appart, tanking the tanks off. anyway. if it helps please let me know, if it doesnt help then sorry, once again im not really sure what your issue is. just kinda sounds like the ski runs for a few seconds then dies when its in the water. thats exactly what happens to my buddies skis and then they get pissed. BTW i have yet to get the skis to run PRIMO, but they will function half decently with the carb adjustments iv provided.