The Shopify Advanced plan allows you to override your store's shipping rates with Printful's live rates. You can also use this feature for other Shopify plans by contacting Shopify and requesting they add live rates for an extra $20 per month.
You can enable live rates automatically by clicking "Override Shopify shipping settings with Printful shipping prices" in your Shopify integration settings which you will find under: Apps > Printful > Settings > Carriers & Services. That checkbox looks like this:
Please check this box only if all of the items on your shop are fulfilled by Printful. If there's at least one item that Printful doesn't recognize (you haven't added the print file and product for its variation), Shopify will use your own default shipping settings and not the real pricing from Printful.
This will only override your own manual shipping rates if you have set any. You will want to ensure that carrier-calculated rates have been disabled with Shopify.
If you do plan to offer products that are not fulfilled by Printful some paid apps like Advanced Shipping Rules will allow you to calculate "blended shipping rates" - meaning you'll be able to generate live shipping rates for both Printful products and the ones you fulfill in another method. Check out their app in the marketplace here.
Instructions for using Advanced Shipping Rules with Printful can be found here as well.