Prove the edges of a multigraph may be oriented such that the net-degree of any vertex is $\leq 1$.
For any graph $G$, perhaps you could add an auxiliary vertex $x$ and make it adjacent to all vertices of odd degree. Call this new graph $G'$. Since there must be an even number of such vertices (so as to keep the degree sum even), $G'$ has an Eulerian circuit. Assign the orientation to $G'$ realizing the circuit starting from $x$. Removing $x$ yields an orientation for $G$ where for each vertex $v\in V(G)$, $netdeg(v)$ is 0 if $deg(v)$ is even and $netdeg(v)$ is 1 if $deg(v)$ is odd.