Soft manipulators exhibit coupled geometric-material nonlinearities under large deflections, which complicate modeling, shape estimation, and closed-loop force control while incurring high computational costs. In this paper, a control-oriented reduced-order adaptive piecewise Euler-Bernoulli (APEB) beam model for tendon-driven slender soft arms is presented. A nonlocal end-force correction is explicitly embedded in the bending equilibrium to capture the projection and attenuation of distal tendon tension and external loads toward the proximal end. To curb the cost of high-dimensional discretization, we introduce curvature/strain-gradient-guided adaptive segmentation with local cubic-spline interpolation: nodes are refined only in regions of severe deformation, whereas fewer degrees of freedom (DoFs) are maintained elsewhere. Based on this discretization, we derive the complete weak-form residual, assemble the tangent stiffness (Jacobian) matrix, and formulate an iterative Newton update, which yields a solver that efficiently computes static equilibria with numerically stable convergence. The model enables fast skeleton reconstruction (shape estimation without external optical tracking) and direct end-effector Jacobian generation through local-to-global coordinate mapping, thereby mapping tendon inputs to the end-effector pose for closed-loop and force control. Simulations and benchtop experiments on a four-tendon continuum actuator show millimeter-level accuracy in the trunk shape and end-effector position across varying cable tensions, and the proposed approach outperforms conventional kinematic and classical beam-theory models in terms of both accuracy and computational efficiency. The model is intended for quasistatic, bending-dominated configurations in which torsional deformation and out-of-plane loading are negligible.