singular shock solutions for strictly hyperbolic system of conservation law

I have no idea if your claims are true, and I am not a specialist of the two systems above. Nevertheless, one can make the following observations.

For the first system ("shallow water"), the eigenvalues of the flux's Jacobian matrix are $\lambda_\pm = q \pm \sqrt{\rho + q^2}$. Therefore the system is strictly hyperbolic if $\rho + q^2 > 0$. The gradient $ \nabla \lambda_\pm = \pm\left(\tfrac12, \lambda_\pm \right)^\top / \sqrt{\rho + q^2} $ of the eigenvalue $\lambda_\pm$ is never orthogonal to the corresponding right eigenvector $(-\lambda_\mp/\rho, 1)^\top$ over the domain of hyperbolicity. Therefore, both characteristic fields are genuinely nonlinear, and the Riemann solution is a combination of shocks and rarefaction waves.

For the second system ("Chaplign gas equation"), the eigenvalues are $(q\pm 1)/\rho$. Therefore, the system is strictly hyperbolic if $0 \neq |\rho| < +\infty$. The gradient $ \nabla \lambda_\pm = \left(-\lambda_\pm, 1 \right)^\top /\rho $ of the eigenvalue $\lambda_\pm$ is always orthogonal to the corresponding right eigenvector $\left(1/\lambda_\pm, 1 \right)^\top$ over the domain of hyperbolicity. Therefore, both characteristic fields are linearly degenerate. You may find the article (1) interesting, which may present a similar problem.

(1) H. Cheng, "Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type", Adv. Math. Phys. (2013), 958120 doi:10.1155/2013/958120