  We carry out a systematic study of the $1S$-wave hidden and double
charm-strange tetraquarks $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ in a
nonrelativistic potential quark model framework with the explicitly correlated
Gaussian method, and the mass spectra, color-spin configurations and possible
decay modes are obtained. We find that although these states are all above
their open flavor thresholds, their rearrangement decay widths are rather
narrow which can be understood by the mismatching of the wave functions between
the initial and final states. It implies that the tetraquarks of
$cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ may have a good chance to exist as
genuine tetraquark states. It also shows that the color-spin configurations of
the $cs\bar{c}\bar{s}$ and $cc\bar{s}\bar{s}$ systems are quite different. We
find that for a physical state of $cs\bar{c}\bar{s}$ its color configurations
can be dominated by either the $|11\rangle_{c}$ or $|88\rangle_{c}$ ones. It
suggests that some hidden charm-strange tetraquark states may strongly couple
to two color-singlet hadrons if the kinematics and dynamics allow. In contrast,
we find that the color configurations $|11\rangle_{c}$ and $|88\rangle_{c}$ in
a double charm-strange $cc\bar{s}\bar{s}$ state are rather compatible. It may
suggest that an overall color-singlet tetraquark (i.e. a genuine color-singlet)
should always play a role in the $T_{cc\bar{s}\bar{s}}$ states. Discussions
taking into account some experimental candidates are presented, and suggestions
on further experimental searches are also made.
