In my experience, you are correct about the AH-4 or the 7000 not maintaining the tuned settings once the 7000 is powered off and the AH-4 gets its power from the 7000 so it powers off as well,
The power problem was an oxidized power connection. I use Anderson Powerpole connectors on all my equipment for universal connectivity and when I initially reconnected the 7000 after sitting effectively idle for over three years, it worked on two nets. Then I connected the HF side and the power problems started so I didn't immediately think it was a power problem.
I'm still working/researching the tuner/analyzer issue.
Thanks for your response,