Don't worry about the gas unless you are going to carry 5 gallons or more when using the generator. If you do carry that amount, mix the Stabil into the gas can. ISTR, the EX650 has a metal tank. If so keep the tank filled when stored. If it has a plastic tank, you can run it dry or keep it full, your choice.
Either E-10 gas is blended differently in different regions of the US or it is just bad on the West Coast. Here, in the Midwest, we've had it for years and nobody seeks any special gas for their small engines. For a few years, I maintained a fleet of small engines (*125) and all were stored for more than 6 mos. with E10 gas and Stabilizer. I didn't have any problems from the gas. During the six mos of summer use, all operated on E10 (no stabilizer) with no problem.