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A number of proposals have been put forward for detecting axion dark matter (DM) with grand unification scale decay constants that rely on the conversion of coherent DM axions to oscillating magnetic fields in the presence of static, laboratory magnetic fields. Crucially, such experiments $\unicode{x2013}$ including ABRACADABRA $\unicode{x2013}$ have to-date worked in the limit that the axion Compton wavelength is larger than the size of the experiment, which allows one to take a magnetoquasistatic (MQS) approach to modeling the axion signal. We use finite element methods to solve the coupled axion-electromagnetism equations of motion without assuming the MQS approximation. We show that the MQS approximation becomes a poor approximation at frequencies two orders of magnitude lower than the naive MQS limit. Radiation losses diminish the quality factor of an otherwise high-$Q$ resonant readout circuit, though this may be mitigated through shielding and minimizing lossy materials. Additionally, self-resonances associated with the detector geometry change the reactive properties of the pickup system, leading to two generic features beyond MQS: there are frequencies that require an inductive rather than capacitive tuning to maintain resonance, and the detector itself becomes a multi-pole resonator at high frequencies. Accounting for these features, competitive sensitivity to the axion-photon coupling may be extended well beyond the naive MQS limit.