Extending the approach of Grillakis, Shatah, and Strauss, Bronski, Johnson, and Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem $$ \min_{X:C(X)=c_0} \mathcal{E}(X) , \ \ \ c_0\in \mathbb R^r , $$ and stability of solutions of a class of degenerate ``quasi-gradient'' systems $dX/dt=-M(X)\nabla \mathcal{E}(X)$ admitting constraints, including Cahn--Hilliard equations, one- and multi-dimensional viscoelasticity, and coupled conservation-law--reaction-diffusion systems arising in chemotaxis and related settings. Using the relation between variational stability and the signature of $\partial c/\partial \omega \in \mathbb R^{r\times r}$, where $c(\omega)=C(X^*_\omega)\in \mathbb R ^r$ denote the values of the imposed constraints and $\omega\in \mathbb R^r$ the associated Lagrange multipliers at a critical point $X^*_\omega$, we obtain as in the Hamiltonian case a general criterion for co-periodic stability of periodic waves, illuminating and extending a number of previous results obtained by direct Evans-function techniques. More interestingly, comparing the form of the Jacobian arising in the co-periodic theory to Jacobians arising in the formal Whitham equations associated with modulation, we recover and substantially generalize a previously mysterious ``modulational dichotomy'' observed in special cases by Oh--Zumbrun and Howard, showing that co-periodic and sideband stability are incompatible. In particular, we both illuminate and extend to general viscosity/strain-gradient effects and multidimensional deformations the result of Oh-Zumbrun of universal modulational instability of periodic solutions of the equations of viscoelasticity with strain-gradient effects, considered as functions on the whole line. Likewise, we generalize to multi-dimensions corresponding results of Howard on periodic solutions of Cahn--Hilliard equations.