It doesn't matter that Barney, Floyd and Opie all witness Elizabeth Crowley (Jean Hagen) speeding through Mayberry on her way back to Washington, D.C. By the time Andy is ready to hear the case, she has charmed all of the witnesses into not testifying against her.
She eventually learns that her methods are wrong and pays her proper fine. But she never backs down from her claim that Barney bears an "amazing resemblance to Frank Sinatra."