The prototypical exciton model of two interacting Dirac particles in graphene was analysed by Sabio et al, Phys. Rev. B 81, 045428 (2010), and it was found that in one of the electron-hole scattering channels the total kinetic energy vanishes, resulting in a singular behaviour. We show that this singularity can be removed by extending the quasiparticle dispersion, thus breaking the symmetry between upper and lower Dirac cones. The dynamics of an electron-electron pair are then mapped onto that of a single particle with negative mass and anisotropic dispersion. We show that the interplay between dispersion and repulsive interaction can result in the formation of bound, Cooper-pair-like, metastable states in double-layered hybrid structures. We analyse these states by calculating their binding energies, decay rates into the free- electron continuum and semiclassical trajectories. We also analyse the problem of bi-electron pairing with the inclusion of the two dominant many-body effects at zero temperature: screening of the Coulomb interaction by the Dirac sea, and reduction of the available phase space due to Pauli blocking of transitions into the states below the Fermi level. We show that these effects result in strong renormalization of the binding energy, but do not destroy the metastable states. Thus the binding energies are strongly dependent on the chemical potential owing to the combined effects of screening and Pauli blocking. Hence, the quasibound resonances can be tuned by electrostatic doping.