We report on our lattice QCD study of coupled $DD_s^*$–$D^*D_s$ scattering in the $J^{P}=1^{+}$ channel and elastic $DD_s$ scattering in the $J^{P}=0^{+}$ channel, aimed at investigating the possible existence of $cc\bar{u}\bar{s}$ tetraquarks near threshold. The calculation uses CLS ensembles with $m_{\pi}\sim280,\mathrm{MeV}$, lattice spacing $a\sim0.09,\mathrm{fm}$, and spatial extents $L/a=24,32$. Finite-volume spectra are obtained from a variational analysis of two-point correlation matrices constructed from two-meson operator bases using distillation. The $l=0$ partial-wave scattering amplitudes are determined from the lattice spectra in multiple moving frames using Lüscher's formalism as well as a finite-volume implementation of the Lippmann–Schwinger equation. In both channels we observe small but nonzero shifts relative to the noninteracting spectrum, indicating weak meson interactions. The extracted physically plausible $S$-wave amplitudes show no pole structures near threshold.