Physics-informed deep learning (PIDL) surrogates can replace traditional numerical PDE solvers and thus accelerate the solution of PDE-constrained optimization (PDECO) problems. However, such surrogates exhibit prediction errors in the state variables and corresponding gradients with respect to the controls, which can lead to nonoptimal or even infeasible solutions of the underlying PDECO problem. To overcome this limitation of static PIDL surrogates, we propose a new approach that combines the concepts of the trust region method with adaptive PIDL surrogates. The PIDL surrogate is first trained on the entire control space to capture general physics. The surrogate is then refined locally around the current candidate point using a transfer learning method, keeping the retraining cost low. This locally refined surrogate is used to solve the PDECO problem with one of the classical gradient-based algorithms. The new candidate point is evaluated using a criterion similar to that of the standard trust region method but using only the surrogate models rather than the costly numerical solution. The process is repeated until convergence, taking into account a small update between candidate points and sufficient accuracy of the current local surrogate. Furthermore, we also use Sobolev-type conditioning to improve the gradient estimates, which is crucial for the gradient-based optimization algorithms used in PDECO. Finally, the effectiveness and efficiency of the proposed approach is demonstrated using various PDEs and PDECO problems. Since we currently use a conditional PINN as the PIDL surrogate, we only showcase PDECO problems with low-dimensional controls. However, the approach can be easily extended to classical PDECO problems with high-dimensional controls by employing more advanced DL architectures such as physics-informed neural operators.