You might need to do both. Adjusting the claw might be enough to get the trem back to level. Adding a spring is going to effect the stiffness/feel of the trem.
if you are changing from 10's to 9's, it means that there is gonna be less tension. Therefore it is recommended to loosen the claw a bit. i dont think you need to put an extra spring.
I suggest that you loosen the screws (counter clockwise) with minimal turns. if compared to a clock, usually you would want to turn it let's say from 12 o clock to 11:45. do this and tune your guitar. then adjust again as needed until the bridge is level. usually a quarter turn is enough.