this computer is no longer authorized for apps that are installed on the ipad

I'm trying to set up iTunes on a new computer and I've attempted to backup my iPad Mini but I keep getting this error.

I have deauthorized all computers and reauthorized 2 computers and I still have the same problem.

As I walk through the backup, I first get this message:
"There are apps on 'iPad Mini' which are not in your iTunes library. Would you like to back up these apps from your iPad?

These apps will not restore on a device when you restore, unless they are in your iTunes library."

I click on "Back up apps."

Syncing begins with preparing to sync (step 1 of 4) and progresses to the back up (step 2 of 4). As it begins transferring purchases, I get this error:
"This computer is no longer authorized for apps on the iPad 'iPad Mini'. Would you like to authorize this computer for items purchased from the iTunes Store?
If you do not authorize this computer, 27 apps and their data will be deleted from this iPad."

I click "Authorize"....even though I have already deauthorized all computers and reauthorized this one prior to trying to back up the iPad.

I'm prompted to enter my user ID and password, which I do. I click "Authorize" again and I get this message:
"You cannot authorize more than 5 computers with this Apple ID. To authorize this computer, you mus deauthorize one of the other computers.

Clearly, I'm frustrated. I have never had 5 computers authorized on my account but after the first time I ran into this error, I deauthorized ALL (3) of the computers and reauthorized 2 of them.

iPad Mini (1) operating IOS 9.1 with a 16GB capacity and 2.3 GB still available. My computer is a 64 bit Windows 10 Acer Aspire E 15 Touch with a I5 Processor and 8GB of memory. I have only one Apple ID and all apps have been purchased and installed with this ID.

What is my next step? I'm trying to backup this iPad so I can restore it a new iPad and I don't want to proceed until I can get all the apps I've paid for.

A related question.....can I find a list of the 27 apps that are not going to transfer? Perhaps they are all free apps and I can dump and reinstall them?