We propose and study a dissatisfied adaptive snowdrift game with a payoff parameter r that incorporates a cost for rewiring a connection. An agent, facing adverse local environment, may switch action without a cost or rewire an existing link with a cost a so as to attain a better competing environment. Detailed numerical simulations reveal nontrivial and nonmonotonic dependence of the frequency of cooperation and the densities of different types of links on a and r. A theory that treats the cooperative and noncooperative agents separately and accounts for spatial correlation up to neighboring agents is formulated. The theory gives results that are in good agreement with simulations. The frequency of cooperation f_{C} is enhanced (suppressed) at high rewiring cost relative to that at low rewiring cost when r is small (large). For a given value of r, there exists a critical value of the rewiring cost below which the system evolves into a phase of frozen dynamics with isolated noncooperative agents segregated from a cluster of cooperative agents, and above which the system evolves into a connected population of mixed actions with continual dynamics. The phase boundary on the a-r phase space that separates the two phases with distinct structural, population and dynamical properties is mapped out. The phase diagram reveals that, as a general feature, for small r (small a), the disconnected and segregated phase can survive over a wider range of a(r).