Is it common in English to refuse to drink offered alcohol by saying "I am sorry, but I can't drink. I am behind the wheel today" meaning that today they still need to drive, so they can't drink. I've looked through different cases of "behind the wheel" given by Google and discovered that they all have slightly different connotation - they put emphasis on being in control of something ("Are you behind the wheel in this system or should I ask someone else?", "Who is the one behind the wheel here?")
No, it's not a matter of connotation or emphasis. The distinction is whether or not you're driving a vehicle.
- Driving a vehicle (and thus literally sitting behind the steering wheel). Don't worry, Katie's behind the wheel, and she hasn't been drinking tonight. I'm so excited to finally be behind the wheel of my own car!
- In charge. With Jim behind the wheel, I'm not sure this team will have the leadership it needs.
Unless that something is a car, the examples you gave of "begin in control of something" are being used in the sense of definition 2.
I am sorry, but I can't drink. I am behind the wheel today.
Is that common? I'm not sure, but it sounds perfectly fine to me.
I would probably say
I am sorry, but I can't drink.
- I'm driving today/later/etc. (← keeping it simple)
- I'm the designated driver.
But again, your way is also understandable.