iTunes won't transfer purchases

I moved my iTunes library to a new computer as I'm decomissioning the failing old one. When I try to sync my iPhone I get a dialog:

"iTunes has found purchased items on the iPhone xxx that are not present in your iTunes library. Do you wish to transfer these items from this iPhone back to your iTunes library?"

When I click Transfer I get another message:

"You cannot authorize more than 5 computers."

As I've migrated from computer to computer over the years I failed to deauthorize old computers. So I chose the option on my account page to deauthorize all computers. I then uninstalled iTunes, reinstalled it and reauthorized my new computer. My account page now shows 1 computer authorized. However I still get the "You cannot authorize more than 5 computers" error when I try and transfer my purchases.

Does anyone have an idea of how to resolve this or know of third party software that will transfer my purchases? The only other option iTunes offers is to delete my apps from the iPhone so at the moment I'm unable to sync, update playlists etc.

Thanks for any help with this.

Michael