NASA's Cooperative Autonomous Distributed Robotic Exploration (CADRE) mission, slated for flight to the Moon's Reiner Gamma region in 2025/2026, is designed to demonstrate multi-agent autonomous exploration of the Lunar surface and sub-surface. A team of three robots and a base station will autonomously explore a region near the lander, collecting the data required for 3D reconstruction of the surface with no human input; and then autonomously perform distributed sensing with multi-static ground penetrating radars (GPR), driving in formation while performing coordinated radar soundings to create a map of the subsurface. At the core of CADRE's software architecture is a novel autonomous, distributed planning, scheduling, and execution (PS&E) system. The system coordinates the robots' activities, planning and executing tasks that require multiple robots' participation while ensuring that each individual robot's thermal and power resources stay within prescribed bounds, and respecting ground-prescribed sleep-wake cycles. The system uses a centralized-planning, distributed-execution paradigm, and a leader election mechanism ensures robustness to failures of individual agents. In this paper, we describe the architecture of CADRE's PS&E system; discuss its design rationale; and report on verification and validation (V&V) testing of the system on CADRE's hardware in preparation for deployment on the Moon.