Your K-1 visa expired the day you entered the US. Your I-94 is the document that shows your legal presence in the US for 90 days following entry.
Used to be you needed to wait a couple of weeks to be sure your name showed up in the SAVE system so that SS could verify your presence. Now that should happen within a day or so. You have, I believe, 76 days following entry to get your SSN or after that time you will need to wait until you have your work document or green card.
The problem that most people have that wait until after marriage is a name change. If you wait until after marriage and change your name, that new name will not show up in their search. I would imagine that even if you were married if you applied for a SSN, did not mention marriage, and used the name that is on the I-94, that you could still get a SSN. It would mean a return to the SS office later to change the card to the new name.