As I see it, this could only be caused by a defect in the Auto-Turn-Off circuit in the combination switch--a very expensive part. It is that device's job to remove power from the tail relay coil by removing ground from the Red-Green wire at T6 of plug C29 when ground is removed from T1 of the same plug by the stalk switch.
This is off a 2002 wire diagram. I assume a '97 has the auto-turn-off feature, if not, disregard this post.
I don't know of any sure way to fix it other than swapping the Combination switch (aka Gauge Cluster, if buying used), or at least the circuit board (if buying new).
You could try pulling the cluster, tracing the circuit from that T6 and look for a burnt trace or some short you could fix but it's long odds. (Again, a '97 may be different as to terminal and wire color--need the '97 EWD) but the function should be there. Obviously, this is just an internet guess and I would be more confident if I could put my test meter to it myself with a '97 EWD in hand. It is more to suggest a direction for your thinking.