© 2016 American Physical Society. We present measurements of Ω and φ production at midrapidity from Au+Au collisions at nucleon-nucleon center-of-mass energies sNN=7.7, 11.5, 19.6, 27, and 39 GeV by the STAR experiment at the BNL Relativistic Heavy Ion Collider (RHIC). Motivated by the coalescence formation mechanism for these strange hadrons, we study the ratios of N(Ω-+Ω#x0002B;)/[2N(φ)]. These ratios as a function of transverse momentum pT fall on a consistent trend at high collision energies, but start to show deviations in peripheral collisions at sNN=19.6, 27, and 39 GeV, and in central collisions at 11.5 GeV in the intermediate pT region of 2.4-3.6 GeV/c. We further evaluate empirically the strange quark pT distributions at hadronization by studying the Ω/φ ratios scaled by the number of constituent quarks (NCQ). The NCQ-scaled Ω/φ ratios show a suppression of strange quark production in central collisions at 11.5 GeV compared to sNN≥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.