Lisa here from Shopify Support!
It appears your store may still have the USPS generated rates which are showing, additionally to the Standard Shipping which you have manually added.
To remove USPS rates, from your Shopify Admin, go to Settings > Shipping. Next Scroll down to the Zone you wish to remove the rates from click 'Edit'. Once in the Zone, scroll down to 'Real-time Rates' and press the 'x' next to this section. Save your change in settings, and it will just display the Free + Weights you've set up.
If you have any other issues with this, please feel free to connect with us via Email, Phone or Chat!
- Lisa ?
Were you able to tackle this problem? Also i am curious to know how are you managing free shipping? Is Free shipping available for all products, or do you offer it after an order amount?