To reduce CO2 emissions and tackle increasing fuel costs, the aviation industry is swiftly moving towards the electrification of aircraft. From the viewpoint of systems and control, a key challenge brought by this transition corresponds to the management and safe operation of the propulsion system's onboard electrical power distribution network. In this work, for a series-hybrid-electric propulsion system, we propose a distributed adaptive controller for regulating the voltage of a DC bus that energizes the electricity-based propulsion system. The proposed controller -- whose design is based on principles of back-stepping, adaptive, and passivity-based control techniques -- also enables the proportional sharing of the electric load among multiple converter-interfaced sources, which reduces the likelihood of over-stressing individual sources. Compared to existing control strategies, our method ensures stable, convergent, and accurate voltage regulation and load-sharing even if the effects of power lines of unknown resistances and inductances are considered. The performance of the proposed control scheme is experimentally validated and compared to state-of-the-art controllers in a power hardware-in-the-loop (PHIL) environment.