  We present JWST NIRCam imaging targeting 13 $z\sim3$ infrared-luminous
($L_{\rm IR}\sim5\times10^{12}L_{\odot}$) galaxies from the ALESS survey with
uniquely deep, high-resolution (0.08$''$$-$0.16$''$) ALMA 870$\mu$m imaging.
The 2.0$-$4.4$\mu$m (observed frame) NIRCam imaging reveals the rest-frame
near-infrared stellar emission in these submillimeter-selected galaxies (SMGs)
at the same (sub-)kpc resolution as the 870$\mu$m dust continuum. The newly
revealed stellar morphologies show striking similarities with the dust
continuum morphologies at 870$\mu$m, with the centers and position angles
agreeing for most sources, clearly illustrating that the spatial offsets
reported previously between the 870$\mu$m and HST morphologies were due to
strong differential dust obscuration. The F444W sizes are 78$\pm$21% larger
than those measured at 870$\mu$m, in contrast to recent results from
hydrodynamical simulations that predict larger 870$\mu$m sizes. We report
evidence for significant dust obscuration in F444W for the highest-redshift
sources, emphasizing the importance of longer-wavelength MIRI imaging. The
majority of the sources show evidence that they are undergoing
mergers/interactions, including tidal tails/plumes -- some of which are also
detected at 870$\mu$m. We find a clear correlation between NIRCam colors and
870$\mu$m surface brightness on $\sim$1 kpc scales, indicating that the
galaxies are primarily red due to dust -- not stellar age -- and we show that
the dust structure on $\sim$kpc-scales is broadly similar to that in nearby
galaxies. Finally, we find no strong stellar bars in the rest-frame
near-infrared, suggesting the extended bar-like features seen at 870$\mu$m are
highly obscured and/or gas-dominated structures that are likely early
precursors to significant bulge growth.
