Post-merger galaxies are unique laboratories to study the triggering and interplay of star formation and active galactic nucleus (AGN) activity. Combining new, high-resolution Jansky Very Large Array (VLA) observations with archival radio surveys, we have examined the radio properties of 28 spheroidal post-merger galaxies. We detect 18 radio sources in our post-merger sample and find a general lack of extended emission at (sub)kiloparsec scales, indicating the prevalence of compact, nuclear radio emission in these post-merger galaxies, with the majority (16/18; 89%) characterized as low luminosity. Using multiwavelength data, we determine the origin of the radio emission, discovering 15 new radio AGNs and three radio sources likely associated with star-forming (SF) processes. Among the radio AGNs, almost all are low luminosity (13/15; 87%), inconsistent with a relativistic jet origin. We discover a new dual AGN (DAGN) candidate, J1511+0417, and investigate the radio properties of the DAGN candidate J0843+3549. Five of these radio AGNs are hosted by a SF or SF-AGN composite emission-line galaxy, suggesting that radio AGN activity may be present during periods of SF activity in post-mergers. The low-power jets and compact morphologies of these radio AGNs also point to a scenario in which AGN feedback may be efficient in this sample of post-mergers. Lastly, we present simulated, multifrequency observations of the 15 radio AGNs with the Very Long Baseline Array and the very-long-baseline interferometry capabilities of the Next-Generation VLA to assess the feasibility of these instruments in searches for supermassive black hole binaries.