Sunny here from the Shopify Guru team. I would be happy to look into this for you.
Just to clarify, are you printing your shipping labels directly from the Shopify admin, or are you using a third pard app? If you are printing them from within Shopify, there are a few areas in the admin where addresses are stored. Let's go through each of those areas to make sure we cover our bases.
Take a look through each of those areas and let me know what you find. If you're not 100% sure and want a second pair of eyes, feel free to send us an email at firstname.lastname@example.org, pass along the ticket number in this thread and I would be happy you take a closer look.
I look forward to hearing from you!
Sunny | Shopify Guru
In case anyone comes up against the same issue - we had our new address correct in all three settings areas (General, Locations, and Shipping), but we didn't Deactivate the original location in "Locations". That solved the issue.