Yes, that is the safest route.
Assuming you still have Captivate 7 installed on the old laptop, select Help > SignOut or Help > D
Close Captivate, and you may want to relaunch it to be sure you get a 'please sign in' or 'please register' message.
Then install CP7 on the new laptop using the same Adobe ID.
I hope you have the CP7 installer still