We report a Karl G. Jansky Very Large Array (JVLA) search for redshifted CO(1-0) or CO(2-1) emission, and a Hubble Space Telescope Wide Field Camera~3 (HST-WFC3) search for rest-frame near-ultraviolet (NUV) stellar emission, from seven HI-selected galaxies associated with high-metallicity ([M/H]~$geq -1.3$) damped Ly$alpha$ absorbers (DLAs) at $zapprox 4$. The galaxies were earlier identified by ALMA imaging of their [CII]~158$mu$m emission. We also used the JVLA to search for CO(2-1) emission from the field of a low-metallicity ([M/H]~$=-2.47$) DLA at $zapprox 4.8$. No statistically significant CO emission is detected from any of the galaxies, yielding upper limits of $M_{mol}<(7.4 - 17.9)times 10^{10}times (alpha_{CO}/4.36) M_odot$ on their molecular gas mass. We detect rest-frame NUV emission from four of the seven [CII]~158$mu$m-emitting galaxies, the first detections of the stellar continuum from HI-selected galaxies at $zgtrsim 4$. The HST-WFC3 images yield typical sizes of the stellar continua of $approx 2-4$~kpc and inferred dust-unobscured star-formation rates (SFRs) of $approx 5.0-17.5 M_odot$/yr, consistent with, or slightly lower than, the total SFRs estimated from the far-infrared (FIR) luminosity. We further stacked the CO(2-1) emission signals of six [CII]~158$mu$m-emitting galaxies in the image plane. Our non-detection of CO(2-1) emission in the stacked image yields the limit $M_{mol}<4.1 times 10^{10}times (alpha_{CO}/4.36) M_odot$ on the average molecular gas mass of the six galaxies. Our molecular gas mass estimates and NUV SFR estimates in HI-selected galaxies at $zapprox 4$ are consistent with those of main-sequence galaxies with similar [CII]~158$mu$m and FIR luminosities at similar redshifts. However, the NUV emission in the HI-selected galaxies appears more extended than that in main-sequence galaxies at similar redshifts.
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