I had a similar problem with my 1973 MGB. I say similar because my strange
electrical gremlins also appeared when I shifted. From memory, my problem
was caused by the seat belt warning light system that was activated when the
car was shifted out of neutral. My advice would be to check all of the
electrical switches that operate when the transmission is shifted. Reverse
lights, overdrive, seat belt ( not sure your earlier 71 has this) warning
I traced the problem by copying the electrical schematic and enlarging it on a
copy machine to 17 inch copy. I then traced each malfunctioning circuit by
highliting that circuit with a different colored marker. I believe at one
point I found a wire common to all of the malfunctioning circuits.
Eventually I found a bad diode located behind the radio. The diode was
allowing current to flow both ways.
But the symptoms only occurred under the right conditions. For example, the
shift lever position, whether the seat belts were buckled and if the door was
open and the courtesy light was on.
I hope this helps.