And I just checked MW on Onelook:

errant from "a road";

errand from "business or a message"

...and the words from which errant and errant were each derived look very different from each other.

MW defines errand in the second sense as the object of a trip. OED defines it in the second sense as object of a journey (the concise edition). And I think what tsuwm shows above explains it--the errand was originally something more involved than the simple errands we think of running today--probably originally more involved with the business world, such as it was.

Oh, well. Much ado about.