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Underwater acoustic localization (UWAL) is a key element in most underwater communication applications. The absence of GPS as well as the signal propagation environment makes UWAL similar to indoor localization. However, UWAL poses additional challenges. The propagation speed varies with depth, temperature, and salinity, anchor and unlocalized (UL) nodes cannot be assumed time-synchronized, and nodes are constantly moving due to ocean currents or self-motion. Taking these specific features of UWAL into account, in this paper, we describe a new sequential algorithm for joint time-synchronization and localization for underwater networks. The algorithm is based on packet exchanges between anchor and UL nodes, makes use of directional navigation systems employed in nodes to obtain accurate short-term motion estimates, and exploits the permanent motion of nodes. Our solution also allows self-evaluation of the localization accuracy. Using simulations, we compare our algorithm to two benchmark localization methods as well as to the Cramér-Rao bound (CBR). The results demonstrate that our algorithm achieves accurate localization using only two anchor nodes and outperforms the benchmark schemes when node synchronization and knowledge of propagation speed are not available. Moreover, we report results of a sea trial where we validated our algorithm in open sea.