In two-sided markets with transferable utility ('assignment games'), we study the dynamics of trade arrangements and price adjustments as agents from the two market sides stochastically match, break up, and re-match in their pursuit of better opportunities. The underlying model of individual adjustments is based on the behavioral theories of adaptive learning and aspiration adjustment. Dynamics induced by this model converge to approximately optimal and stable market outcomes, but this convergence may be (exponentially) slow. We introduce the notion of a 'market sentiment' that governs which of the two market sides is temporarily more or less amenable to price adjustments, and show that such a feature may significantly speed up convergence.