Hey Sarah!
Thanks for reaching out. I tried out the checkout page on your site and I did see the issue just as you described. The option to enter a shipping address would only appear if I selected Australia as the Billing country.
That’s not how the system is designed to work. I test on my development site by setting my sell-by to all countries and the shipping countries to only be Australia. At checkout the ship to checkbox was present regardless of the billing country. I could use a billing address from anywhere, but only ship to Australia.
Here are my shipping settings to see how they compare to yours:


You can see if any changes to these make a difference. If not, then I would recommend a conflict test to see what’s preventing this from working as designed. You may want to do that on a staging site to keep from disrupting your live site.
Temporarily switch to the Twenty Nineteen theme. Then disable all plugins except for WooCommerce. See if you can check out and select a shipping address at checkout. If you can then turn your theme back on. Then try again. Continue testing by enabling plugins. When it stops working, the last item activated will likely be the source of the trouble.
Let me know how it goes and we’ll take it from there.
Cheers