I am going to jump back in here. I have not replied again yet as I don't know this particular brand or model of cart. I am going to make some observations and maybe assumptions.
A fully charged battery will read 13.6-13.8 volts after a full charge. Either your battery is bad, the charge circuit is not fully charging it, your meter is not reading correctly or you have a bad connection at the battery posts.
Give the battery another full charge and read the voltage directly on the battery posts without it connected to the cart. If you still only get 11.* volts, we need to address this first before checking the wiring on the cart.
Now I need some clarification from you. When you are checking each side of the solenoid and seeing way less volts on the generator side, is this with the engine running or not? If the engine is running and the charge circuit is working, you should see around 14.5 volts at the battery and both terminals of the solenoid.
You need to see 13.* volts at the battery before pressing the pedal, then when you push the pedal, you should see the voltage drop to around 10/11 volts and then rise to 14.5 volts or more once the engine starts and revs rise enough to energise the starter to become a generator. You should see these voltages at all the points I mentioned previously. If not, we need to find out why.
You ask about the green wire. I cannot comment on your particular cart, but generally a thin green wire at the starter/generator will be connected to the voltage regulator to control rate of charge back to the battery. Ideally, I need to see a wiring diagram of your model of cart to further diagnose what is happening here.
The battery voltage at rest is what concerns me at the moment. Maybe it needs to be load tested to confirm it is good and a different meter should be tried to confirm actual battery voltage.
With a six cell lead acid battery I would expect a minimum of 13.2 volts at rest a few hours after charging either with a proper wall charger or charging from the cart being driven.