How to supply power to a 12V solenoid from China, getting hot
If the coil resistance is 9 ohms, it should draw 1.33 amps at 12 volts, and dissipate 16 watts - so it will get quite hot.  That current is much more than a common 9 volt battery can supply - from the current you mention, the battery voltage will be about 4.8 volts when trying to drive the valve.

Despite the eBay ad's claim, I suspect it may actually be an AC solenoid.  On one dealer's website, I see coil powers for that size valve stated as 3 - 6 watts.

Adding resistors in series with the coil won't help - it will just reduce the current, and the voltage across the coil.