The first few Macbook Air models had HDDs, (and they were rubbish.)
If you are receiving a new MBA, I would be extremely reluctant to void the warranty by opening it up.
We need to know what year your old MBA is before we can advise further on enclosures. If it uses SSDs, then one solution could be to ask your repair shop to transfer the info onto a flash drive or normal SSD (ie. non-Apple SSD).
You can sell the old Macbook SSD - they are highly specialist items and bring in a fair bit on Ebay. The repair shop might be willing to make you an offer for it.