We report results from a deep search for redshifted HI 21cm absorption from eight damped Lyman-$alpha$ absorbers (DLAs) detected in our earlier optical survey for DLAs towards radio-loud quasars. HI 21cm absorption was detected from the $z = 2.192$ DLA towards TXS2039+187, only the sixth case of such a detection at $z > 2$, while upper limits on the HI 21cm optical depth were obtained in six other DLAs at $z > 2$. Our detection of HI 21cm absorption in the eighth system, at $z = 2.289$ towards TXS0311+430, has been reported earlier. We also present high spatial resolution images of the background quasars at frequencies close to the redshifted HI 21cm line frequency, allowing us to estimate the covering factor of each DLA, and then determine its spin temperature $T_s$. For three non-detections of HI 21cm absorption, we obtain strong lower limits on the spin temperature, $T_s gtrsim 790$ K, similar to the bulk of the high-$z$ DLA population; three other DLAs yield weak lower limits, $T_s > 140-400$ K. However, for the two DLAs with detections of HI 21cm absorption, the derived spin temperatures are both low $T_s = (160 pm 35) times (f/0.35)$ K for the $z = 2.192$ DLA towards TXS2039+187 and $T_s = (72 pm 18) times (f/0.52)$ K for the $z = 2.289$ DLA towards TXS0311+430. These are the first two DLAs at $z > 1$ with $T_s$ values comparable to those obtained in local spiral galaxies. Based on the observed correlations between spin temperature and metallicity and velocity spread and metallicity in DLAs, we suggest that the hosts of the two absorbers are likely to be massive, high-metallicity galaxies.
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