I'm trying to add my card to my account but it keeps trying to send a verification code to an old number I no longer have access to and the website is not letting me continue without the verification how can I delete that number off my account so that I can continue with my process?
Try downloading the app and changing the phone number via settings / using the forgotten password option / trying again another day when sometimes you can get in without doing the verification and then you can go and change the number.
If that does not work then you would need to contact customer services so that they can i.d. you another way and then add your new phone number to your account.
Paypal phones are mostly non operational since covid but sometimes you get lucky. (use the guest option if you can't log in). If there is one available it will show when you click help / contact bottom left of Paypal pages for options available in your country. Live chat is randomly available. You can send them a message. Or have you considered contacting Customer Service via Facebook or Twitter? You can send them a personal message from their facebook or twitter pages. It's: https://www.facebook.com/PayPal and @AskPayPal for Twitter.
Advice is voluntary. Kudos / Solution appreciated.