In this paper we propose a novel method for communication over underwater acoustic channels for which the product of delay-spread and Doppler-spread is large, such as those pertaining to the surf zone. In particular, we propose the use of pulse-shaped multicarrier modulation to convert the doubly dispersive channel into an inter-carrier interference (ICI) channel with small ICI spread. We then propose a novel joint ICI-estimation/data-detection strategy which performs near optimally yet at low complexity, due to the use of an efficient tree search algorithm and the ability to leverage sparseness in the delay-power profile. The delay-power profile, which itself varies in time, can be readily tracked via pilots. Numerical simulations show that our technique gives performance close to genie-aided bounds over highly spread channels.