Stateful Network Functions (NFs) are widely deployed in cloud environments, where operations such as scaling, failover, and migration depend on NF state to ensure correct processing. Supporting these operations requires state transfer between NFs. The emergence of Multi Access Edge Computing (MEC) introduces state transfer scenarios such as state splitting and merging across NFs, driven by user mobility and distributed deployment. Existing techniques perform state transfer at NFs, which becomes a bottleneck, particularly when a NF transfers state to other NFs while continuing packet processing or competing for CPU resources. We present BoostState, a low-latency state transfer framework in MEC that leverages programmable data planes to accelerate state transfer while preserving conventional cloud-based NF deployments. BoostState leverages state partitioning, where NF state is partitioned into header and action state. It delegates handling header state to programmable data planes, reducing state transfer overhead at NFs. We implement BoostState using P4 language on a Netronome SmartNIC. Our results show that BoostState reduces state transfer latency compared to prior work, including up to 60% lower transmission latency.