An investigation of the steep, high-frequency (i.e., ~12 GHz) radio spectra among a sample of 31 local infrared-bright starburst galaxies is carried out in light of their HST-based merger classifications. Radio data covering as many as 10 individual bands allows for spectral indices to be measured over three frequency bins between 0.15-32.5 GHz. Sources having the flattest spectral indices measured at ~2 and 4 GHz, arising from large free-free optical depths among the densest starbursts, appear to be in ongoing through post-stage mergers. The spectral indices measured at higher frequencies (i.e., ~12 GHz) are steepest for sources associated with ongoing mergers in which their nuclei are distinct, but either share a common stellar envelope and/or exhibit tidal tails. These results hold after excluding potential AGN based on their low 6.2µm PAH EQWs. Consequently, the low-, mid-, and high-frequency spectral indices each appear to be sensitive to the exact merger stage. It is additionally shown that ongoing mergers, whose progenitors are still separated and share a common envelope and/or exhibit tidal tails, also exhibit excess radio emission relative to what is expected given the far-infrared/radio correlation, suggesting that there may be a significant amount of radio emission that is not associated with ongoing star formation. The combination of these observations, along with high-resolution radio morphologies, leads to a picture in which the steep high-frequency radio spectral indices and excess radio emission arises from radio continuum bridges and tidal tails that are not associated with star formation, similar to what is observed for so-called "taffy" galaxies. This scenario may also explain the seemingly low far-infrared/radio ratios measured for many high-z submillimeter galaxies, a number of which are merger-driven starbursts.
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