We present new Atacama Large Millimeter/submillimeter Array (ALMA) continuum and NH2D, N2D+, and H2D+ line emission at matched, ∼100 au resolution toward the dense star-forming cores SM1N and N6 within the Ophiuchus molecular cloud. We determine the density and temperature structure of SM1N based on radiative transfer modeling and simulated observations of the multiwavelength continuum emission at 0.8, 2, and 3 mm. We show that SM1N is best fit by either a broken power-law or Plummer-like density profile with high central densities (n ∼ 108 cm−3), and an inner transition radius of only ∼80–300 au. The free-fall time of the inner region is only a few ×103 yr. The continuum modeling rules out the presence of an embedded first hydrostatic core (FHSC) or protostar. SM1N is therefore a dynamically unstable but still starless core. We find that NH2D is likely depleted at high densities within SM1N. The nonthermal velocity dispersions increase from NH2D to N2H+ and H2D+, possibly tracing increasing (but still subsonic) infall speeds at higher densities as predicted by some models of starless core contraction. Toward N6, we confirm the previous ALMA detection of a faint, embedded point source (N6-mm) in 0.8 mm continuum emission. NH2D and N2D+ avoid N6-mm within ∼100 au, while H2D+ is not strongly detected toward N6. The distribution of these tracers is consistent with heating by a young, warm object. N6-mm thus remains one of the best candidate FHSCs detected so far, although its observed (sub)millimeter luminosity remains below predictions for FHSCs.