You don't say what Avast Product that you had ?
You actually needed to cancel it through Avast as they would be the ones submitting the payment request to PayPal.
Orders and Renewals FAQ -
https://www.avast.com/faq.php?article=AVKB24#artTitle. Scroll down to the "How do I cancel automatic renewal of my Avast subscription?" option.
If your PayPal uses your credit card as a payment source, check the 'Bank accounts and cards' in your Summary section. If it appears there then that is how payment is made, but don't be confused, it isn't Avast that has your CC details, that is PayPal debiting your CC the Avast program renewal charge/request.
I don't know where you requested the refund twice as you didn't say ?
But having tried to get a refund via PayPal, that could actually make things more complex.
Before shooting the messenger, I'm an Avast user like yourself.