Here's a guide I've found useful in my experimentation with Shopify stores in the past: https://help.shopify.com/en/manual/shipping/rates-and-methods/manual-rates
Shipping is definitely one of the trickier pieces of configuration. If the guide itself isn't self-explanatory, can you try entering a weighted rate threshold that starts when the flat rate shipping would become cheaper? For example, let's say the flat rate is $8.00 and you start using it when packages are 13 ounces.
If that doesn't work, let me know. I'd like to see if we can troubleshoot this together.