My mom has to use a cane for a few years now and she was embarassed at first also.
When I took her to Disney last year for her first trip ever we actually got an
ECV from an outside vendor. That was a life saver and I tell everyone to get one who asks. You can take it on all of the different forms of transportation to and from the parks to the Hotels and a few of the rides too. The majority of the rides she had to transfer but the whole group went with her through the handicap entrance and there was usually a spot to park the ECV. And, if you were to get off the ride at another place other than where you boarded the cast members would move the vehicle for you. WDW is very handicap friendly, some of the guests may have an issue but just ignore them. If it wasn't for the ECV my mother would not have had a chance of enjoying herself as much as she did. Plus, it gave us a place to store stuff we bought so that we didn't have to carry it all day long.
As far as the plane, my mom would just ask someone to help her out and have the cane placed in the overhead area or she would place it against the window/wall if she was sitting there.