We formulate a source-driven electromagnetic theory for a uniform screw-dislocation Riemann–Cartan medium without reducing the vector Maxwell system to independent scalar transverse equations. The defect geometry and the effective axial electromagnetic response are kept as distinct ingredients, allowing their relation to be specified by a microscopic or phenomenological material coupling. Starting from a gauge-compatible Carroll–Field–Jackiw-type action, we derive the complete harmonic Maxwell equations in the anholonomic cylindrical frame and factorize the source-free problem exactly into two Beltrami sectors. Their longitudinal fields obey confluent-hypergeometric radial equations, while the transverse electric and magnetic components are reconstructed algebraically and satisfy Gauss’s law identically. A distributionally conserved circular shell current then leads to a closed four-amplitude matching problem with explicit Kummer and Tricomi solutions. The same radial operator is mathematically equivalent to a two-dimensional Landau problem: normalizable modes form a Laguerre–Gauss tower and the screw geometry lifts the $m\leftrightarrow -m$ degeneracy even at zero axial electromagnetic coupling. Away from the exceptional Beltrami surfaces, the singular points of the matching matrix reproduce this guided spectrum; the fundamental exceptional branch is recovered directly from the first-order system. For experimental interpretation, the homogeneous local limit is recast in ordinary material language: the two circular eigenwaves acquire distinct effective refractive indices, producing reciprocal optical rotation and polarization conversion. Near the forward optical branch, the guided-level spacing and the splitting between Beltrami sectors provide independent leading-order readouts of the geometric scale $\Og$ and the axial response $v$. The construction also clarifies the role of the axial projection of torsion, the limitations of a uniform timelike effective coupling, and the breaking of standard electromagnetic duality at finite axial response.