I can't really help with the customer service issue. I know it barely exists.
Can you get the addresses from the emails sent when orders come in and then use Pirate Ship for your labels? At least this way, your customers will still receive their orders. Their email addresses should be in those order confirmations as well so that you can communicate with customers. You could send tracking numbers that way.
I wouldn't express in detail to your customers what's going on as far as the Plaid issue. I would maybe just say you're experiencing glitches with the Etsy system at the moment; therefore, you're sending all information (shipping, answering messages, so on and so forth) directly via email.
