Terry, the AH-4 will maintain it's settings after tuning as long as the power is kept on. If you tune pressing the tune button on the IC-7000 and then disconnect the PL-259 on the IC-7000 and connect your analyzer you should be able to check the SWR at that frequency.
The AH-4 should give you a better SWR than 3:1 unless you happen to have a length of wire that is 1/2 wave on the frequency you want or the AH-4 is not working correctly.
Do you have a ground at the AH-4? What is the length of the wire?
The antenna should tune OK on every band if it is 43' or longer, except if it is 1/2 wave resonant.
Are you getting the red tune light on the panel, indicating that the tuner has properly tuned the antenna?