I've been off the boards for a week but I meant to get back to this topic with what I believe is a successful conclusion. I remember Mike Tangas saying to check the connections but they looked OK. It wasn't until I removed the main power cable from the battery that I was actually able to SEE the ignition wire which goes to the solenoid. I reached in and grabbed the wire and to my surprise it was loose. The screw that holds it on was tight but it was still loose. To remedy this I found two very small washers in my toolbox to take up the slack. Once it was tight the problem vanished. Mission accomplished. Check the connections, rinse, repeat.
1967 Jaguar E-type Series I, 2+2