Virginia,
Here are the "gotchas" that you should be aware of:
One: The time limit: The English Heritage Pass (Overseas Visitor Pass) is good for either 9 or 16 consecutive days, depending upon which ticket you buy. It is activated when you visit the first attraction and the card is scanned. This means that once you use it (swipe it), the clock is running. You will, obviously, want to visit one right after another of these attractions, before the clock (time) runs out.
Therefore, look at the attractions listed on a map. Group the ones that are close to one another, hit them by area. You could also list the attractions on the list you'd like to visit, look at which ones are the most expensive, use the Pass on all the most expensive admissions FIRST to maximize your savings.
Two: The card is not replaceable or refundable if you lose it.
Three: The card is also not transferable to another person, in case you cannot take your trip as planned, and you have already bought it ahead of time online. When you use the card the first time to activate it, they will require an ID with your name and home address on it (passport or driver's license). It must match the name who made the purchase online of the Pass.
I do have experience using this Pass and found that I saved money, but you must have a definite plan about which historic palaces and sights you want to see, and hit them quickly. Of course you will not be able to see all the attractions covered by the pass. That's not the point of the pass. You will begin to save money after you have visited the first 6 or 7 sights, in my opinion. Hit your own personal "must see" sights first.
The following sights are covered at Hadrian's Wall: Birdoswald Roman Fort, Chesters Roman Fort, Corbridge Roman Town and Housesteads Roman Fort.
When you return to London, the following sights are covered by the pass: Apsley House, Chiswick House, Eltham Palace and Gardens, the Jewel Tower, Marble Hill House, Wellington Arch, and Ranger’s House (which is in Greenwich).
The sights you have named are spread out in England. Dover and Hadrian's Wall are quite far from one another, as you know. So, you will lose some travel time off the clock between them, but there are some interesting sights on the way from one to the other. Also, your savings at some of the London venues help you feel like you got your money's worth.