That is a very good idea asking them to write the information on the first dose card. Thank you! Worst case scenario could be to get the second dose in the States to be fully vaccinated and then get a third jab in Europe if they cannot validate it but it would be taking a dose from someone else. In Sweden, they now seem to have more vaccines than people interested in getting them so they are making more people eligable faster.
Yes, Florida is one of those States. As we have a condo with an address there, a deed and a utility bill, we tick that box.