Jetski has sat for a year, half a tank of premium gas..should I drain?

No question you should drain it and replace with fresh fuel. You can use the old gas in something with a much bigger like your car or truck. Simply only put a gallon or so each time you fill up.

Drain the carb if possible. But I understand it is running so if you burn the gas off that should be ok.

Don't add new to old. If you decide to burn what you have, save the new gas and put it in after the old is gone. I'd bet there is water at the bottom. How are you going to get that out?

I would just top it off with fresh gas if it ran on the gas that was still in the tank it can't be that bad..

Edit If the gas was bad it wouldn't even run and if it had water in it it wouldn't even start because the water will be in the lowest part of the tank so it would be the first thing to go up the pickup tube. If you want to throw away money go ahead and drain it and then put fresh gas in it. The next thing it's illegal to just dump gas on the ground so you will have to drive to a recycling center and pay them to take it. Also if the gas is not good enough for your jet ski why in the hell would you take a chance of screwing up the motor in your car or truck?