Yes, the free shipping option in the pricing step during product push lets you publish that product on your storefront with free shipping. This option is available for Shopify, Etsy, eBay, Bonanza, and Wish integrations when adding new products to your store.
By switching on the Free shipping toggle button in pricing settings, you can see the end price if shipping is included. Since you’ll have to cover the shipping costs, add them to your product retail price.
Here's how products with free shipping look like on Etsy:
While your customers won't be charged for shipping, you still have to make up for the costs because Printful will charge you. Consider pricing your products a bit higher to earn a profit and keep the free shipping selling point.