Looking at that link I sent you: it looks like it is saying "deauthorize all" is only permitted once a year. If you deauthorize the products on your old computer one at a time, I expect you will be ok. I confess I haven't had to do this in a long time, so I would go slowly and pick one of the products to deauthorize on your old computer, and then install and authorize it on your new computer to make sure the process works ok. If something happens to your new computer in the future and you need help getting things working correctly on a replacement, I expect Fab support staff will assist.
So I suggest you pick one of the products you have installed & authorized on your existing computer, deauthorize it (do not "deauthorize all") and then install/authorize that product on the new one. I am also going to contact Fab support and ask them to look over this message stream to make sure we are handling this correctly.