Are you travelling on a senior Fare?
The Southwest site states
"If you are traveling on a Senior Fare, and you are not age verified, you MUST provide valid proof of age (driver's license or other accepted identification) at the airport when exchanging your confirmation number for a boarding pass. You will not be able to check in online or at a RAPID CHECK-IN Kiosk. A Medicare card is not considered valid proof of age."
They have a process to make yourself eligible for online checkin but the whole process takes 4 - 6 weeks from start to receiving a confirmation letter in the mail once your date of birth has been verified. (ie create a My Southwest Account; make a copy of your drivers license or another government-issued photo identification that indicates your date of birth and mail to Southwest Airlines Customer)
If not travellying on a senior fare, even if you were to get a C boarding pass I doubt you would be bumped from the flight. Once people starting queuing, jump in queue to be near the front of the C queue. When we went in February, I only recall 1 person with a C boarding pass who didn't get to seat with their party. They actually had a better seat because they found a vacant seat towards the front where most of those who have A boarding passes first sit. Also if you have young children under 5 you get to preboard regardless of what letter you have on your boarding pass.