There is a Distribution charge and a Fuel Adjustment charge on my utility bill. Does this mean I am being charged twice for my energy usage?

No, you are not being double charged for your electric usage.  There are two components of your electric bill directly related to the energy you use-the distribution charge and fuel adjustment charge. 

The distribution charge recovers the cost of operating the power plant and maintaining infrastructure such as the transmission and distribution lines that provide electricity to your home.

The fuel adjustment charge only recovers fuel costs, purchased power costs, and fuel related expenditures.  This means whatever the City pays is passed onto the customer as the fuel adjustment charge on your utility bill.