When a colleges says it “meets full need”, it does not mean it meets your need as your family sees it. All colleges that meet full financial need calculate your need using their individual methods. Three different universities might calculate your need as three different amounts.
As mentioned earlier, the FAFSA EFC is not necessarily relevant here. The FAFSA EFC should be renamed Pell Grant & Work Study Calculator. I believe those are the two biggest things affected by the EFC.
Thus, even Smith might fall short of leaving your family with a bill of $13K or less. Have you run the Net Price Calculator at all your daughter’s top colleges? In most circumstances, that can give you a fairly accurate estimate of your COA.