Escorting My Mother

My mother recently retired, and now that she has the time, she wants to travel a lot to visit our out-of-town relatives she rarely got to see before (her other children, who are my siblings, her grandchildren, and her elderly mother, who is my grandmother). Our family is geographically scattered, and to her, this is her opportunity to see them more often.
But there is a hitch. She is not comfortable traveling far from home by herself. She wants someone to accompany her on all these trips by plane, train, and automobile. And that someone just happens to be me because I’m her only child who lives near her.
I am in my 40s, and I am a happily married man with a daughter away at college. My mother is in her 60s and she looks very young and is very energetic. She easily passes for being my age. When I have been with her in public, people have mistaken her for being my wife! I find this so humiliating!
There is no one else around who can escort my mother. My father can’t do it because he works very hard and is extremely busy and doesn’t like to travel. My wife can’t come because she works full time too and also doesn’t like to travel. We have no other relatives living in this area available to do this, and unlike my father and my wife, I do like traveling. This leaves the burden on me. And I am the only one she trusts. She wants me to take trips with her almost every week!
My father insists I accompany my mother on her travels. My wife also thinks I should. My mother is counting on me. I feel torn about this. What should I do?