This paper addresses multi-UAV uniform sweep coverage in an unknown convex environment, where a homogeneous UAV swarm must evenly visit every portion of the environment for a sampling task without access to their position and orientation. Random walk exploration is practical in this scenario because it requires no localization and is easy to implement on swarms. We demonstrate that the Self-Organizing Nervous System (SoNS) framework, which enables a robot swarm to self-organize into a hierarchical ad-hoc communication network using local communication, is a promising control approach for random exploration in such environments. To this end, we propose a SoNS-based random walk method in which UAVs self-organize into a line formation and then perform a random walk to cover the environment while maintaining that formation. We evaluate our approach in simulations against several decentralized random walk strategies. Results show that our SoNS-based random walk achieves full coverage faster and with greater coverage uniformity than these benchmark strategies, both globally and in local regions.