Great question! It sounds like you've created flat shipping rates in your store, but what you're looking to do is charge live carrier rates (the shipping cost that is actually paid).
Depending on how many products you have and whether they vary in size, you could get away with Shopify shipping. This also assumes you're in North America or Australia, Shopify only offers one box size to be sent off for live rate calculations, so if your products vary in size you will be regularly over/undercharging your customers for shipping. If they are consistently sized, you will do great with the single box size.
In the even that your products do vary in size, Smart Boxing would be a fantastic fit to charge accurate live rates to your customers. Our proprietary boxing algorithm virtually 'packs' orders at checkout, returning accurate live shipping rates to your customers. If you're interested, you can book a call with our team here or contact us at firstname.lastname@example.org to learn more.