Yes, I agree with the chaps. Run an errand. That's the usual one.

"Do an errand" is possible, but not that common. I think it's a confusion with "do work". We normally say "run an errand".

I'll be right back, I'm going to run an errand. You don't generally say DO an errand. Though you could say "I have an errand to do" or "I have errands to do" or "I have errands to run."