I presume that your '84 RX7 controls the high/low beams with a stalk mounted on the steering column. From your description, it seems that the electrcial circuits to the headlamps are on order and that the problem is mechanical with the the high-beam position in the switch. It will not hold the "high" position. It's high-low beam toggle is damaged or the switch is loose. Could it be that you are not moving the stalk far enough and engaging only to the "flash high beam" position (provided the '84 had that feature)?

Probably a relay that is opened/closed with the stalk switch..
The economical VW (1980) did not use a relay, but I'll bet the RX does.
Also check the grounds..