In this work, we present a new fractional-step approach for two-phase flows within an adaptive finite element framework, using a level-set formulation. Standard fractional-step schemes rely on extrapolated or estimated pressure fields to achieve second-order accuracy. Moreover, when dealing with two fluids of different densities under the action of gravity, it is convenient to introduce pressure enrichment in order to represent the discontinuity of the pressure gradient across the fluid interface. The combination of these two ingredients raises several issues when they are applied together within a segregated scheme, which we address in the present work. The first contribution of this work is the use of an explicit enrichment function for the pressure field. In contrast to monolithic schemes, the pressure field is solved in a decoupled manner, which implies that an independent estimation of the enrichment function must be used when computing the intermediate velocity field. This is mandatory when dynamic adaptive mesh refinement is employed. To this end, an element-local hydrostatic solution is used to compute the pressure enrichment function. The second contribution addresses the fact that the position of the interface, represented by a level-set function, changes between successive time steps. This requires a careful choice of the configuration at which the various terms in the segregated scheme are evaluated, particularly in the case of extrapolated quantities. Numerical examples are presented to illustrate the performance of the method in both hydrostatic and dynamic configurations.