Wow, this got depressing fast.
I've never really had a chain of recurring dreams, but I have had a series of pseudo-anxiety dreams; typically they're dreams in which I fuck something major up (kill someone/prevent the resurrection of a dead person, lose something valuable) and everyone forgives me and is entirely placid about it. A similar theme has been me being entirely placid about some such thing: "wait, it's finals week and I never even bought any of my textbooks! Ah, it's okay, I think I know most of the material, and I can go to the bookstore if I'm confused."
The one "chase" dream I've had was sort of similar: a group of classmates (not my actual classmates) and I were touring a giant glass skyscraper when we were captured by a serial killer. He very calmly explained everything to us -- he would give us each a Tic-Tac, and then he would kill us -- and we took everything very well until he refused to let me finish my gum before taking the Tic-Tac and dying, at which point I freaked out, attacked him ineffectually, and then ran. The problem was that the entire skyscraper was made of clear glass, so he could see me wherever I ran and could give chase... but somehow I got out scot-free, felt an immense flood of exhilaration and relief, and woke up. It wasn't really a nightmare, even though it should have been, but was just profoundly odd.