Dating with Kids

I seem to be struggling with the dating-with-kids thing. I am lucky that my almost 14 year old is willing to babysit my 9 year old while I am out for dinner. But last night she got freaked out and started texting me, and I had to come home. Yesterday, I was driving with my kids in the car and I saw him on the street. I yelled and he saw me, waving like a lunatic, so I pulled the car over before even really thinking about it. And then, there I was awkwardly having to introduce him to my kids, something I vowed I wouldn't do until it was clear that I wasn't competing with others, and there was something more serious between us, which might never come. So is it possible to be dating where all it amounts to is dinner once a week? The whole thing seems doomed. What do you do, when you want to date, invest time to see where something might lead, but don't have the luxury of an ex to pass your kids off on occasionally?