Transition state (TS) searches are a critical bottleneck in computational studies of chemical reactivity, as accurately capturing complex phenomena like bond breaking and formation events requires repeated evaluations of expensive ab-initio potential energy surfaces (PESs). While numerous algorithms have been developed to locate TSs efficiently, the computational cost of PES evaluations remains a key limitation. In this work, we develop and fine-tune a graph neural network (GNN) PES to accelerate TS searches for organic reactions. Our GNN of choice, SchNet, is first pre-trained on the ANI-1 dataset and subsequently fine-tuned on a small dataset of reactant, product, and TS structures. We integrate this GNN PES into the Freezing String Method (FSM), enabling rapid generation of TS guess geometries. Across a benchmark suite of chemically diverse reactions, our fine-tuned model (GNN-FT) achieves a 100% success rate, locating the reference TSs in all cases while reducing the number of ab-initio calculations by 72% on average compared to conventional DFT-based FSM searches. Fine-tuning reduces GNN-FT errors by orders of magnitude for out-of-distribution cases such as non-covalent interactions, and improves TS-region predictions with comparatively little data. Analysis of transition state geometries and energy errors shows that GNN-FT captures PES along the reaction coordinate with sufficient accuracy to serve as a reliable DFT surrogate. These results demonstrate that modern GNN potentials, when properly trained, can significantly reduce the cost of TS searches and broaden the scope and size of systems considered in chemical reactivity studies.