"but the software tells me the max we can put as a family is $4800. "
Is this from you pressing the "Calculate Max Contribution Limit" button on the screen of type of coverage per month? If so, please ignore it. It does not handle the situation of a married couple sharing Family HDHP coverage well. If this is not what you are referring to, please tell me where TurboTax told you that your limit was $4,800.
Your situation is not the result of any software error, but it will take a few questions to figure out what you did.
You two have Family coverage. What were the months in which both of you were covered by the Family HDHP plan and there was no conflicting coverage (like a health FSA or Medicare)?
Did you make excess contributions to your HSA in 2020? If so, did you carry over the excess to 2021?
Do both of you have an HSA? Are either of you 55+?
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