Headlight / Relay Kit Issue

Check your grounds REALLY well. That is a classic ground issue.

Use a wire and ground them directly to Batt - and see if anything changes. Disconnect the grounds from the current locations and direct ground them, see if that change the situation.

You could also test light the relay blocks with relays removed. Probe the opening with the orange and purple 16 ga wire. With switch off, there should be no power there. With low the purple should be hot, orange will be hot when in high beam mode. It's not a trigger issue from what you describe, both lights would be on, see next paragragh.

Only one headlight staying on is inconsistent with the design of my system. Each headlight has a dedicated wire for both high and low operation. If one side was on, the other should be on as the wires are fed from same relay for hi or low mode.