Paired state of nonstandard quasiparticles is analyzed in detail in two model situations. Namely, we consider the Cooper-pair bound state and the condensed phase of an almost localized Fermi liquid (ALFL) composed of quasiparticles in a narrow-band with the spin-dependent masses (SDM) and an effective field, both introduced earlier and induced by strong electronic correlations. Each of these novel characteristics are calculated in a self-consistent manner. We analyze the bound states as a function of Cooper-pair momentum q in applied magnetic field in the strongly Pauli limiting case (i.e. when the orbital effects of applied magnetic field are disregarded). The spin-direction dependence of the effective mass makes the quasiparticles comprising Cooper pair spin distinguishable in the quantum mechanical sense, whereas the condensed gas of pairs may still be regarded as composed of identical entities. The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) condensed phase of moving pairs is by far more robust in the applied field for the case with spin-dependent masses than in the situation with equal masses of quasiparticles. Relative stability of the Bardeen-Cooper-Schrieffer (BCS) vs. FFLO phase is analyzed in detail on temperature - applied field plane. Although our calculations are carried out for a model situation, we can conclude that the spin-dependent masses should play an important role in stabilizing high-field low-temperature (HFLT) unconventional superconducting phases (FFLO being an instance) in systems such as CeCoIn_5, organic metals, and possibly others.