You're right, and its shape IS distorted. The effect is just so tiny it's not noticeable. It helps to put some numbers on it:

• Our galaxy is ~87,000 light years across. So to an observer, the light from far edge would be 87,000 years older
• But it takes our galaxy 250,000,000 years to complete 1 rotation!! So you're right, there IS a distortion - the light from the back edge appears to be shifted by 0.0003 of a rotation.

The amount it has moved through the universe is just negligible compared to how fast light is compared to stuff.