Luckily our son loves all things transportation, so the airport and airplane were like two big toy stores for him. Not only did he see a line of taxis but he took the elevator, saw many conveyor belts, watched the AirTrain and saw planes taking off. And our daughter was also content. She enjoys being in a front carrier and her eyes widened as she saw all of the airport's and airplane's new sites.
When we boarded the Southwest flight I found two seats for us. Almost immediately a woman nearby took one look at us and quickly stowed away her laptop and other belongings to move as far away from us as she could. It was a full flight and, not surprisingly, the seat next to us wasn't filled until the last moments. Thankfully, a kind man who has three kids himself asked if that seat was available. He even joked about having lots of experience with kids grabbing him and spitting up, so I need not worry. I couldn't have asked for a better seatmate.
I don't know how often I will travel on my own with two kids. But at least now I know that I can handle it.