I know that has been asked a few times in the past, and I see that those threads are all Read-Only - so I'm starting a new one with how do it.
All of those old addresses that you see when you go to order something on a merchant's website are considered "Gift Addresses". They can only be removed by calling customer service.
1) Navigate to: https://www.paypal.com/us/selfhelp/home
2) Click "Call Customer Support"
3) State that you want to manage your GIFT ADDRESSES
Someone should be able to help you super easy, ask for an Account Specialist if you're having trouble.
I talked to a nice woman who, after helping me, suggested that I submit a Feedback form (link found in the bottom right corner on the PayPal Homepage) that stated that this was something the community wanted. She did submit a form from her end, but then also stated that PayPal listens more to the community than to the Specialists. So, I encourage any/everyone reading this to submit that form so that hopefully no one else has to endure this.
