In the present study, we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in a variety of parameter regimes under which the finite-time blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the α-models of turbulence and therefore are called the α-regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the α-regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work [Cao et al., Nonlinearity 21, 879-898 (2008); Cao et al., Numer. Funct. Anal. Optim. 30, 46-69 (2009)] on the non-viscous α-regularization of the nonlinear Schrödinger equation.