When you run the uninstaller, you have two options: 1. Leave the activation info on the computer, so that when you install it again on the same computer, you don't have to activate it again, or 2. Deactivate it, so that you can install it on a different computer and activate it there.
You could choose either option, if you choose option 2 then you'll obviously have to activate it again after installing.
You shouldn't get any license problems, but if you do you can just phone Adobe and explain what you're doing and most likely they'll help you with the licenses.
I don't know if re-installing is going to solve the problem... Have a look in the help files if there's a way that you can enable the missing option.