See if I'm getting it: the correct use is "avoiding the question," as you said, "begging OFF the question."
Also, in spite of what it sounds like, "begging the question" doesn't necessarily involve an actual question, right? Like in parliamentary procedure, you say "call the question," which isn't about inquiry-- it means put an end to debate and settle the issue. So begging the question really means evading the issue?
The incorrect usage is sort of like "skipped investigation of the question and went right to a likely/obvious conclusion."
