We consider a class of convex Nash games where strategy sets are coupled across agents through a common constraint and payoff functions are linked via a scaled congestion cost metric. A solution to a related variational inequality problem provides a set of Nash equilibria characterized by common Lagrange multipliers for shared constraints. While this variational problem may be characterized by a non-monotone map, it is shown to admit solutions, even in the absence of restrictive compactness assumptions on strategy sets. Additionally, we show that the equilibrium is locally unique both in the primal space as well as in the larger primal-dual space. The existence statements can be generalized to accommodate a piece-wise-smooth congestion metric while affine restrictions, surprisingly, lead to both existence and global uniqueness guarantees. In the second part of the technical note, we discuss distributed computation of such equilibria in monotone regimes via a distributed iterative Tikhonov regularization (ITR) scheme. Application to a class of networked rate allocation games suggests that the ITR schemes perform better than their two-timescale counterparts.