While Orthogonal Frequency Division Multiplexing (OFDM) has been studied for a decade, the study of Non-Orthogonal Frequency Division Multiplexing (NOFDM) in underwater acoustics (UWA) communication has never been reported, to the best of our knowledge. Therefore, we investigate NOFDM technique for UWA communication considering the doubly-dispersive channel. The main feature of NOFDM is that it involves more closely packed subcarriers compared to OFDM, which results in higher spectral efficiency. However, this transmission suffers from severe Inter-Carrier-Interference (ICI). Since, we are considering doubly-dispersive channel, Inter-Symbol-Interference (ISI) is also encountered due to multipath propagation. Therefore, traditional receiver of NOFDM system in UWA channel, tends to have high computational complexity. Considering this problem, we design a receiver for NOFDM system, where Basis Expansion Model (BEM) is used along with Compressed Sensing (CS) channel estimation technique i.e., Orthogonal Matching Pursuit (OMP), which can effectively reduce the computational complexity. However, while implementing this technique in the real sea environment, the problem of long delay / Doppler spread is encountered due to guard intervals. Therefore, we propose the use of time domain equalizer, to mitigate the effect of long delay / Doppler spread. Simulation and experimental results based on Bit Error Rate (BER) demonstrate performance degradation due to severe interference in our proposed receiver. On the contrary, Mean Square Error (MSE) performance shows that our proposed receiver with OMP channel estimation outperforms OFDM receiver. Similarly, higher spectral efficiency gain is attained due to closely packed subcarriers.