This paper proposes a hybrid control methodology which integrates feedback and switching for constrained stabilization of parabolic partial differential equation (PDE) systems for which the spectrum of the spatial differential operator can be partitioned into a finite slow set and an infinite stable fast complement. Galerkin's method is initially used to derive a finite-dimensional system (set of ordinary differential equations (ODEs) in time) that captures the dominant dynamics of the PDE system. This ODE system is then used as the basis for the integrated synthesis, via Lyapunov techniques, of a stabilizing nonlinear feedback controller together with a switching law that orchestrates the switching between the admissible control actuator configurations, in a way that respects input constraints, accommodates inherently conflicting control objectives, and guarantees closed-loop stability. Precise conditions that guarantee stability of the constrained closed-loop PDE system under switching are provided. The proposed methodology is successfully applied to stabilize an unstable steady-state of a diffusion-reaction process using switching between three different control actuator configurations.