Okay. Let me clear this up for you.
If both systems have a HDD, then recovering the gamertag is enough to have it on both consoles. You can never be online with the same gamertag at the same time. However, you can have it on both HDD and login into it if you'll only be using either one once at a time while online.
The only reason you would need a memory unit is if one of the 360s had no HDD. That doesn't seem like the problem in your case since saves would also become a problem.