Physics-based Earth system models (ESMs) are essential for attributing climate change and generating scenario projections, yet their reliance on high-resolution numerical integration makes multi-decadal experiments expensive. In parallel, deep learning has delivered strong gains in short-range weather forecasting; however, auto-regressive roll-outs can accumulate error and become unstable when extended to decade-scale climate emulation. We introduce A-UTE: Advection Informed, Uncertainty Aware Temperature Emulator, aimed at stable multi-year emulation across heterogeneous climate models and grid resolutions. A-UTE is trained on various physics-based models at varying spatial resolutions to emulate temperature fields over a 10-year horizon. A-UTE formulates climate emulation as a forward-time stochastic dynamical system. We propose an auto-regressive ODE-SDE surrogate in which transport dynamics are constrained by an advection consistent ODE component, while a learned neural SDE term models coarse-grained variability and cross-model discrepancy at monthly resolution. We train A-UTE under negative log-likelihood objective for principled uncertainty estimates and probabilistic evaluation. Experiments across 20 climate models show that A-UTE improves long roll-out stability and accuracy relative to relevant baselines, advancing data-driven climate emulation with explicit physical structure and uncertainty-aware predictions.