  We present measurements of $\Omega$ and $\phi$ production at mid-rapidity
from Au+Au collisions at nucleon-nucleon center-of-mass energies
$\sqrt{s_{NN}}$ = 7.7, 11.5, 19.6, 27 and 39 GeV by the STAR experiment at the
Relativistic Heavy Ion Collider (RHIC). Motivated by the coalescence formation
mechanism for these strange hadrons, we study the ratios of
$N(\Omega^{-}+\Omega^{+})/(2N(\phi))$. These ratios as a function of transverse
momentum ($p_T$) fall on a consistent trend at high collision energies, but
start to show deviations in peripheral collisions at $\sqrt{s_{NN}}$ = 19.6, 27
and 39 GeV, and in central collisions at 11.5 GeV in the intermediate $p_T$
region of 2.4-3.6 GeV/c. We further evaluate empirically the strange quark
$p_T$ distributions at hadronization by studying the $\Omega/\phi$ ratios
scaled by the number of constituent quarks. The NCQ-scaled $\Omega/\phi$ ratios
show a suppression of strange quark production in central collisions at 11.5
GeV compared to $\sqrt{s_{NN}} >= 19.6$ GeV. The shapes of the presumably
thermal strange quark distributions in 0-60% most central collisions at 7.7 GeV
show significant deviations from those in 0-10% most central collisions at
higher energies. These features suggest that there is likely a change of the
underlying strange quark dynamics in the transition from quark-matter to
hadronic matter at collision energies below 19.6 GeV.
