This is Rae from Shopify. I'm happy to help look into this!
To clarify, have you already updated your shipping policy in the 'Settings > Legal' section of your Shopify admin? If so, once you make your changes here and click 'Save' at the bottom of the page, these edits should be reflected on your live site, too. However, if they aren't, you may need to clear your browser cache to have the new information show up correctly.
If this doesn't resolve things though, I recommend double checking how your shipping policy is set up on your site. For instance, if you created this policy in the 'Online store > Pages' section of your admin, rather than in the 'Legal' section, you'll need to update your page with the new details in order for the changes to appear on your live site.
Is your shipping policy part of one of the menus on your site? If so, it's worth verifying that the policy that you're editing is linked to your menu correctly. To do this, just head into the 'Online store > Navigation' section of your Shopify admin, and click on the menu where your shipping policy is added. Then, click 'Edit' beside your shipping policy to make sure it's linked to the policy you're editing. Make sure to click 'Apply changes' when you're done linking the correct policy here.
Let me know if these steps resolve things, and if your policy is showing up on your live site with your edits. However, if you still can't see the changes, just reply back to me here so I can continue helping out with this.
In the meantime though, I wanted to also share this blog post with you, as it offers some great tips on what should be included in a shipping policy, and it also outlines some example policies that you might find helpful while editing your own policy.
I hope to hear back from you about this soon!