US Immigration from Ukraine

Hi guys,

So, apparently I can already file N-400 - married to USC for 4+ years and LPR since 30 Aug 2017. After getting married, I changed my last name on everything I could think of, starting with my SSN card - so it's hyphenated with my maiden name and my husband's last name. When my 2-year GC arrived, they'd dropped the hyphen (I hadn't caught it in the interview for that card). We then moved to a different state, and the local DMV used my GC non-hyphenated last name for the new DL. I haven't changed any of the long-term things (like banking info, SSN card) because I was going to have my name fixed on the 10-year GC anyway.

I've just received my 10-year GC with my last name corrected to include the hyphen. So basically I now have my GC, SSN card and the majority of other stuff (like debit/credit cards) carrying my correct last name, but my current DL has the last name with the error. I've scheduled an appointment at the DMV to get a new DL, but the earliest slot I was able to grab was like the last week of October.

My question is, do I understand correctly that first I should wait for my new DL with correct last name because that would be my legal name? It's just that in the Part 1 of the N-400 form question 1 reads "Your Current Legal Name", and then question 2 is "Your Name Exactly As It Appears on Your Permanent Resident Card", and then the instructions explain that as "Your current legal name is the name on your birth certificate unless it changed after birth by a legal action such as a marriage or court order." The state where we got married doesn't ask "what would you like your name to be now", so the most official thing for the name change that I have is my SSN card.

Hopefully my question makes sense Thanks!