Continuum robots are flexible, slender manipulators well suited for confined surgical environments. In these settings, unknown interaction forces and model uncertainty significantly affect robot shape, motivating state estimation from external observations. Existing estimation methods either neglect actuation modeling or rely on simplified deterministic actuation models. In contrast, we jointly estimate robot shape, external loads, and actuation inputs using mechanically principled actuation priors. To achieve this, we present a discrete Cosserat rod formulation with piecewise-linear strain integration that provides high numerical accuracy while inducing a sparse factor graph structure for efficient nonlinear optimization. We extend the framework to tendon-driven and parallel robots in simulation and validate it experimentally on a surgical concentric tube robot. Overall, our approach enables principled real-time estimation across multiple robot architectures while providing direct access to manipulator Jacobians through the linearized factor graph.