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We theoretically investigated strong-field XUV pump- IR probe dissociative ionization in full dimensionality by propagating the coupled nuclear motion on the A²Π_u, B²Σ_u⁺, C²Σ_g⁺, a⁴Σ_g^-, and b⁴Π_u of CO₂⁺ in full (3D) dimensionality. Our ab initio calculations are based on adiabatic Born-Oppenheimer (BO) potential-energy surfaces and non-adiabatic, laser-dipole, and fine-structure couplings between BO states we obtained by applying the multi-configurational self-consistent-field quantum-chemistry code GAMESS. This allowed us to provide kinetic energy release (KER) spectra for dissociation into the in the O(³P_g) + CO⁺( X²Σ⁺) and O⁺(⁴S_u) + CO(X¹Σ⁺) channels, the relative importance of the mentioned couplings, and branching ratios for O⁺ versus CO⁺ yields for different laser parameters and initial molecular orientations to the linearly polarized laser electric field. Our KER spectra reveal the ro-vibrational excitations of CO⁺ fragments along a dominant 3ω dissociation paths. Addressing the nuclear dynamics near the conically intersecting A²Π_u and B²Σ_u⁺ states, we reproduce the core-hole oscillation period 115 fs in good agreement with the experiment of Timmers et al., Phys. Rev. Lett. 113, 113003 (2014). In addition, we predict and interpret as due to quantum beats between specific vibration and electronic states slower 62 fs oscillation.