Correcting FNU\LNU name issues before applying for green card


My name is Ranjani Srinivasan. I have issues with different names on my passport and US visa.
Passport: Given Name- Srinivasan Ranjani; Surname - Blank
Visa: Given name- FNU; Surname- Srinivasan Ranjani

I am currently in California on H1B and I understand that the process is to contact the indian consulate in my state to change the name in the passport.
Question 1: In my case, I not only have to split my name but I also need to reverse my name from Srinivasan Ranjani to Ranjani Srinivasan. Is this considered a name change and do I have to go through the newpaper ads in India and US?
2: I’m currently on H1B visa and I will be applying for a green card through my husband who is a US citizen. In this case, can I use my marriage certificate in the US which has my correct name as Ranjani Srinivasan. Will this documentation suffice for a name change in the US to apply for green card without correcting my Indian passport?

Also, my birth certificate has my name as S. Ranjani ( the initials were expanded as Srinivasan Ranjani in my passport Given name )

If you have to change the name or order, it maybe considered a major change, need to check with Indian embassy on the exact requirement.

Yes, you may use that, probably need affidavit or notarized…Talk to an attorney, it is better to get their opinion as Green Card is a critical thing and you do not want to make mistake.