As a US citizen, you have up to 90 days to stay here, visa free, thus the visitor stamp on your passport. Home Office have been known to process these permits on a slow pace, and worst case scenario, you might just run out of time.

Yes you do have a right to stay here being the spouse of an EU citizen, and that will be your main argument in discussions with the Home Office. After a month, you can start calling them to follow up (try not to mention the date you gave the application to see what response you get). You can also write them a letter requesting your passport back for urgent travel, and worst case scenario (and hopefully you don't have to take this route), you and hubby leave UK, and then just re-enter and you have a fresh 90 days again.

Nice to see a fellow American on UK soil! Good luck.

 

For big name employers who have proper HR departments, they would very well know that your right to employment stems from the fact you are married to an EU National. The residence card only confirms your right of residence (which also confers with it your right to employment). There are many pages in the Home Office website that will tell you this, which you can print off and show to the HR department.