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What you're describing sounds like you may not have weight conditions set up in your settings. It's a good idea to rectify this to ensure you don't lose money on shipping. In order to charge the correct shipping rate to a specific weight, you will need to set up conditions for weight-based shipping rates. Feel free to follow the steps below to do this.
Repeat steps 4-9 to create other weight conditions with different shipping rates.
Modifying your shipping settings is a good opportunity to rethink your shipping strategy. Keep in mind that increasing the shipping rates may discourage customers from placing an order; and while you're not taking a loss on shipping with that setup, you may be taking a loss on conversions. Make sure you consider different shipping strategies before you decide on one. This blog post shares three shipping tactics that increase conversions. Get some ideas from there so you can determine what shipping setup will be ideal for both you and your customers.