Have you checked the kick-down switch? I've seen it happen where the kick down switch is stuck in and that will prevent OD from engaging. If it is stuck, a drop or two of oil on the spindle should fix it. Otherwise, I don't know if the Darrin is the same, it likely is, as the other KF cars but there are four wires on the white connector; two are thick 12 gauge and two are thin 16 gauge. When plugged into the relay, the front most wire is a the thick orange one (if it has the original harness, the wires likely have faded to the point where color is not distinguishable) and it goes directly to 6 volts, probably the hot side of starter solenoid (the one that connects directly to the neg battery cable.) The wire to the right of the right (when viewing from the front) is the other 12 gauge wire (originally blue) which goes directly to the number 4 terminal on the solenoid. The rear most wire is the white 16 gauge that goes to the normally closed contacts on the kick down switch. The wire on the right, is the other 16 gauge and it goes to the hot side of the coil.
If you take the relay off, shake it gently and see if it rattles. If it does, then the terminal "tower" inside has corroded loose. You can usually fix it if you remove the cover and drill out the two brass rivets that hold the terminal "tower" to the base and replace them with nuts/bolts (making sure not to break the wire to it) and that will likely (not definitely) fix it.
If it doesn't rattle, a good quick test is touch a 9 volt radio battery to the left and rear terminals (as viewed from the front) and it should click (doesn't matter which way you touch the battery, either way will work.) If it does click, that's a good sign that works so the next thing to do would be clean the fuse socket and the brass terminals on the relay and try it again.
Also, here's some other discussions we've had on the topic:
http://kfclub.com/forum/index.php/topic,3996.msg17152.html#msg17152
