SSA replacement card through AOS or Work Authorization forms

I entered the U.S. on a K1 visa and successfully applied for a Social Security number (SSN) through the SSA office, receiving it without 'Work Authorization' noted. Now that I'm married, I need to update my SSN with my new last name. However, my I-94 has expired, and I understand that I'll need a work permit to make this change.

The SSA office is quite far from my home, requiring a 4-hour ride with no public transportation options, so I rely on my husband to drive me there. Unfortunately, due to his new job, it's challenging for us to make this trip, especially since we anticipate needing time off for USCIS biometrics and interviews in the coming months.

While filling out the Adjustment of Status (AOS) forms, I noticed an option to request a replacement SSN card. There's also a similar option in the Work Authorization form. My question is: can I use these forms to update my SSN with my new last name and avoid a trip to the SSA office? Should I select the option on both forms, or is it sufficient to choose it on just one? We're looking for the most efficient way to handle this without needing to take additional time off.