There are different rules for people with EU/EEA driving licences and those with licences from outside the EU (such as the US).
As your husband is Swedish (and presumably had an EU driving licence), he was allowed to go in person to the Croydon office and exchange his Swedish licence for a full UK licence. He would not have been required to get a provisional licence or take any of the UK driving tests... he could just exchange his EU licence for a UK one.
However, as a US citizen (presumably with a US driving licence?) you cannot exchange your licence. You need to apply by post for a UK provisional licence (you keep your US licence if you have one) and then you have to go through the entire process of taking the UK theory and practical driving tests. I don't believe you can apply in person because you are not a UK or EU citizen and unfortunately I believe you have no choice but to send your passport.
If you have been in the UK for more than 12 months (looking at your earlier posts, you've been here since 2008?), you can no longer drive on your US licence if you have one, so you will be treated as a learner driver. This means that you must put 'L' plates on the car, you cannot drive on motorways, and you cannot drive alone and must be accompanied at all times by someone over 21 who has held a UK licence for at least 3 years.