  We present deep GMRT 21cm absorption spectra of 10 damped Lyman-$\alpha$
systems (DLAs), of which 8 are at redshifts $z \ga 1.3$. HI absorption was
detected in only one DLA, the $z = 0.5318$ absorber toward PKS 1629+12, which
has been identified with a luminous spiral galaxy; the spin temperature limit
($T_s \le 310$ K) derived from our observations continues the trend of DLAs
associated with bright spirals having low spin temperatures. In 7 of the
remaining 9 systems, the observations place strong lower limits on the spin
temperature of the HI gas.
  The sample of DLAs searched for 21cm absorption now consists of 31 systems,
with $T_s$ estimates available in 24 cases; of these, 16 are at $z < 2$ and 8
at $z > 2$, with 11 (all at $z < 1$) having optical IDs. For the latter 11
systems, we find that all low $T_s$ DLAs have been identified with luminous
galaxies, while all high $T_s$ ($T_s \ga 1000$ K) DLAs have been identified
with either LSBs or dwarfs. DLA spin temperatures thus appear to correlate with
galaxy type, with no correlation seen between $T_s$ and impact parameter.
  The trend that low $z$ DLAs exhibit both high and low $T_s$ values while high
redshift ($z \ga 3$) DLAs only show high spin temperatures is present in this
expanded data set. Based on this difference in spin temperatures, the Gehan
test rules out the hypothesis that DLAs at $z > 2$ and DLAs at $z < 2$ are
drawn from the same parent population at ~ 99 % confidence level.
  Finally, we estimate upper limits on the fraction of cold HI, $f_{CNM}$, in
the $z \ga 3$ DLAs. In local spirals, $f_{CNM} \sim 0.5$; in contrast, we find
that $f_{CNM} < 0.3$ in all 7 high $z$ DLAs, with $f_{CNM} < 0.1$ in 5 of the 7
cases. (abridged)
