Before you activate your software on another computer, you must deactivate it on the first machine. So, if you still have your first computer with you, I suggest that you install the software on that computer and then, use Help > Deactivate to deactivate your computer. Uninstall the software on your second computer, reinstall, and choose Help > Activate (or wherever the activate button is present on your version of Dreamweaver).
If none of this works, look at other solutions on the Activation Deactivation document. This document is chat enabled, and a chat window appears as soon as a support person is available. I am not sure about Adobe's policy to support a version that is as old as yours, but give it a shot nevertheless.
Not to complicate things, but I believe that this is not necessarily true. For as long as I can remember, you have been allowed to have two copies of DW installed and activated provided that you are not using them both simultaneously (in other words, you are covered for workstation and laptop with a single purchase). But the procedure that you describe is the correct one to follow:
1. Deactivate on old computer
2. Install on new one
3. Activate on new one
As I said... I already deactivated and tried to activate... with no success. Activation methodologies on the old product are not available and not supported.
I called upon Adobe support via Chat, and their support was OUTSTANDING! quite eye opening to the point they certainly earned my next round of business -- Yes, they got my new install actviated by generating a new key - Thanks Adobe!