This paper studies stability conditions and composite stabilization schemes for a class of stochastic hybrid systems. Such systems switch among a finite number of linear singularly perturbed system models, with switched behaviour governed by a nearly completely decomposable finite state Markov chain. The stability properties of a decoupled slow mode subsystem, a decoupled fast mode subsystem and an overall system are also investigated. Linear switching stabilization schemes with perfect as well as imperfect Markov chain detectors are developed to stabilize the overall system. Two examples are provided to illustrate the stabilization schemes.