We consider a source node deployed in a real-time monitoring application that needs to sample a stochastic process and convey its state timely and accurately to a destination over a wireless ON/OFF channel. The source can either process a raw sample to determine its current state and transmit that information or transmit the raw sample and let the destination determine the state. The source is subjected to an average cost constraint, and it cannot sample, process, and transmit at all the time instants due to the associated costs. When the destination does not receive information, it uses the previous information as an estimate of the current state, which, if it matches the actual state at the source, the distortion is considered to be zero. The objective is to minimize average expected distortion subject to constraints on the average expected age of information (AoI) of states of interest and costs incurred by the source, where the AoI of a state increases if no status update is received, else drops to unity. We derive a stationary randomized policy (SRP) to solve the formulated problem, for which we obtain the expression for the expected AoI under the SRP using a lumpability argument on the two-dimensional discrete-time Markov chain formed using AoI and instantaneous distortion as states. We extensively study the impact of the system parameters on the average distortion under the SRP and draw significant conclusions.