The nonlocal kinetic linear stability analysis of the non-Harris thin current sheet equilibrium, namely the thin current sheet embedded in a thicker anisotropic plasma sheet <Sitnov et al., 2000a, 2000b>, with respect to current-driven instabilities is performed using the finite element technique. In contrast to the Harris sheet, the new equilibrium becomes possible due to the plasma anisotropy outside the sheet caused by two warm counterstreaming field-aligned beams and complex ion orbits that cannot be described in such thin current sheets in terms of the conventional magnetic moment. It is found that in contrast to the case of the Harris sheet, the analogs of the drift-kink instability in these current sheets can have significant growth rates for the realistic ion-to-electron mass and temperature ratios. The unstable modes share the properties with both the lower-hybrid and drift-kink modes. In particular, the unstable modes resemble the lower-hybrid drift modes as they are more highly structured across the sheet than the drift-kink instability (DKI) and assume both odd (DKI-like) and even parity solutions. On the other hand, in contrast to the lower-hybrid drift instability (LHDI) and like the DKI, the unstable modes have much larger wavelength, electromagnetic component, and significantly perturb the central current region. The possible role of the current-driven instabilities in magnetic reconnection and magnetic annihilation as well as the geophysical implications such as the current disruption in the geomagnetotail during substorms are also discussed.