Is Dun / dunning archaic?
You ask if the expression is archaic.
Well the OED has an example from 2014, from the New York Times.
2014 N.Y. Times (Nexis) 7 Dec. (Business section) 3 I am now being dunned for $80.63. (Hide quotations)
Not sure what your criteria are for archaic, but clearly it is not something that belongs entirely to an earlier century.