Quantum critical theories with continuously varying critical exponents remain challenging to access experimentally. Here, we propose quantum-dot architectures for realizing the quantum Ashkin-Teller and XYZ/eight-vertex models using resonator-mediated and direct Coulomb interactions, respectively. We focus on the Ashkin-Teller and eight-vertex critical lines, which are connected by a non-local duality relating local order parameters to topological string operators. The resulting platforms provide microscopic control over the parameters explicitly controlling critical exponents, with the Coulomb-based implementation enabling stronger interaction regimes. We demonstrate that continuously varying critical behavior can already be resolved in short chains. For experimentally realistic parameters, the accessible interaction range is expected to produce changes in the critical exponents of order 10% for the Ashkin-Teller model and substantially larger changes for the eight-vertex model. In both cases, the predicted variations lie well above the estimated measurement uncertainties.