Many years ago, I had an issue in one Shop only ( I have 5) which only affected Australian Buyers using the Pay Pal Option at Checkout. Payments failed to process or cancel and this went on for Months. Support could not see the issue and any escalation to the "specialized team" was, similarly, inadequate. It became evident that the US Support reps could not even see the issue as it only affected Buyers logged into Etsy from Australia. Eventually, the issue disappeared and I never knew what had caused it in the first place.
All I can suggest is open an email ticket and keep replying to any automated response with "I still need help". Eventually a real Support person will take up the issue. From experience, I have found it better to keep your emails brief and to the point. That is all you can do.
As far as the Chat disconnecting when you asked for an email to be sent to an alternative address, I see this as a good thind. So many Shops have been hacked through email and fraudulent address changes that Etsy is and should be very relustant to comply with a request like this. Surely you can receive emails sent to the address on file associated with this Shop?
You can also access Chat through Shop Manager > Help > Contact us > Selling on Etsy. You may have to test options and look for the "I still need help" button at the bottom of each page, beneath the suggested solutions
Email is open here
https://help.etsy.com/hc/en-us/requests/new?segment=selling#issue_account_ato_email-contact_select
