Classical resistive tearing theory assumes a prescribed current-sheet equilibrium, but a finite normal magnetic field can evolve the background on the tearing timescale. For a transversely pressure-balanced Harris profile of initial half-thickness $a$ and asymptotic reconnecting-field magnitude $B_0$, threaded by uniform $B_n$ ($\xi \equiv B_n/B_0$), we derive an exact evolution in ideal magnetohydrodynamics: counter-propagating Alfv'enic fronts broaden the neutral layer and generate a sheet-parallel flow with velocity gradients on its two flanks. The same normal field enters the linearized perturbation equations directly through ideal transport, alongside advection and flow-curvature terms from the evolving velocity profile. Inner-layer estimates recover $\xi_{\rm crit}\sim S^{-3/4}$ for the breakdown of the resistive balance at the classical fastest-growing scale, where $S$ is the Lundquist number based on $a$. Numerical frozen-profile calculations show that growing modes can persist after central resistive dominance is lost; eigenfunction comparisons at fixed wavenumber and predominantly Doppler-shifted propagation support a normal-field-modified continuation of classical tearing. The normal field supplies no net perturbation energy. Energy extraction from the reconnecting-field gradient dominates at small and moderate widths and weakens as the sheet broadens. The velocity-gradient contribution becomes comparable only in broad, weakened modes, where it partly offsets the loss without reversing the decline in growth. For the representative case $S={10}^6$, magnetic Prandtl number ${Pr}_m=0$, and $\xi ={10}^{-4}$, the maximum growth rate falls by a factor of about ten from its initial value when the neutral-layer half-width $w$ reaches $w/a\simeq 1.59$, where the growth and broadening times become equal. Only about $0.18$ additional upper-envelope e-folds are estimated between this crossing and $w/a=2$, where the growth rate is about 25 times smaller than its initial value and the instantaneous spectrum remains unstable. These results support dynamical suppression of further linear amplification over the sampled evolution, while nonlinear island or plasmoid formation remains unresolved. Current-sheet disruption criteria must therefore account for both direct modification of the inner balance and normal-field-driven background evolution.