Context. Previous kinematic analysis of the pc-scale molecular torus in NGC 1068 have revealed a very complex and inhomogeneous system involving several physical components, such as non-circular motions, turbulence, high-velocity outflows, and possibly even counter-rotation. Aims. Our study aims to dissect the kinematics and morphology of the molecular gas within the near-nuclear region of NGC 1068 to understand the mechanisms in the central Active Galactic Nucleus (AGN) that might fuel it, and the impact of its energy output on the surrounding molecular gas. Methods. We present high angular and spectral resolution ALMA observations of the HCO+ (J = 4→3) and CO (J = 3→2) molecular lines in the near-nuclear region of the prototype Seyfert 2 galaxy NGC 1068. The spatial resolution (1.1 pc) is almost two times better than that of previous studies examining the same molecular lines at the same transitions and is the highest resolution achievable with ALMA at these frequencies. Our analysis focuses on moment maps, position-velocity (PV) diagrams, and spectra obtained at the position of the nuclear continuum source, along with a simple kinematic model developed using the 3DBarolo software. Results. Our observations reveal significant asymmetry between the eastern and western sides of the nuclear disc in terms of morphology, velocity, and line intensity. The broad lines (σ∼90 km/s) seen in the inner 2 pc could be accounted for by either beam smearing or highly turbulent gas in this region. Outside this radius, the mean velocities drop to ±30 km/s, which cannot be explained by asymmetric drift. We find low velocity connections extending to 13 pc, suggesting interactions with larger scale structures. The CO/HCO+ line ratio at the nucleus reported here are extremely low compared to values in the literature of the same galaxy at lower spatial resolutions. We find high-velocity redshifted clouds in absorption and emission at the nuclear position. Conclusions. The molecular environment near the nucleus of NGC 1068 is highly disturbed and asymmetric, marked by the presence of a high-velocity infalling cloud. High excitation temperatures, high molecular column densities, along with the unusually low CO/HCO+ line ratio close to the nucleus seem to indicate intense interaction with AGN radiation. These findings underscore the complexity of AGN feeding mechanisms and the pivotal role of high-resolution studies in unravelling the physical processes at play near supermassive black holes.