We present relativistic many-body calculations of atomic and molecular Schiff-moment interaction constants including interelectron correlation effects using atomic Gaussian basis sets specifically optimized for the Schiff interaction. Our present best results employing a Gaussian nuclear density function are ${\alpha }_{\text{SM}}=(0.362\pm 0.025)\times {10}^{-17}\frac{e\text{cm}}{e{\text{fm}}^3}$ for atomic ${}^{129}\mathrm{Xe}, {\alpha }_{\text{SM}}=(-2.26\pm 0.23)\times {10}^{-17}\frac{e\text{cm}}{e{\text{fm}}^3}$ for atomic ${}^{199}\mathrm{Hg}$, and $W_{\text{SM}}=(39967\pm 3600)$ a.u. for the thallium nucleus in the molecule ${}^{205}\mathrm{TlF}$. We discuss agreements and discrepancies between our present results and those from earlier calculations on the atoms ${}^{129}\mathrm{Xe}$ and ${}^{199}\mathrm{Hg}$. Using the most recent measurements of $\mathcal{P},\mathcal{T}$-odd electric dipole moments and the present interaction constants, reliable upper bounds on the Schiff moments of the ${}^{199}\mathrm{Hg}$ and ${}^{205}\mathrm{Tl}$ nuclei are determined in the context of a single-source assumption.