  In this paper and its sequel \cite{DPP}, we investigate the precise
relationship between the quadratic affine Weyl group multiple Dirichlet series
in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead
\cite{White2} and \cite{White1}. In particular, we show that the axiomatic
quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle
4}^{\scriptscriptstyle (1)}$ over rational function fields of odd
characteristic admits meromorphic continuation to the interior of the
corresponding complexified Tits cone. We shall also determine the polar divisor
of this function, and compute the residue at each of its poles. As a
consequence, we obtain an \emph{exact} formula for a weighted 4-th moment of
quadratic Dirichlet $L$-functions over rational function fields; we shall also
derive an asymptotic formula for this weighted moment that is expected to
generalize to any global field.
