luggage – Re-dropping off claimed baggage after international flight layover at SFO

I’ll be travelling to Austin from Surat (India) via San Francisco.

The itinerary is
Surat <–> New Delhi <–> San Francisco <–> Austin

The first leg of the flight Surat <–> New Delhi is Air India. While the remaining two legs are on United Airlines (on separate tickets). I have a twelve-hour layover in San Francisco.

I wanted to know whether I can drop/checkin the baggage I pick up after landing in SFO immediately for my next flight so I can venture out in the city – without being burdened with carrying the bags everywhere.

From these two questions on the forum, it seems like a “baggage drop-off” counter exists at the SFO international arrival, however, this would only be a possibility if the bags are already marked for Austin. Which I’m not sure they would be since I booked the two United flights separately.

So my questions finally are:

  1. Will my baggage already be marked for Austin. If not, can they be (since it’s the same airline)?
  2. Where and how do I request for my bags to be marked for Austin so I can avail this immediate drop-off at SFO? In Surat? In Delhi?

If I cannot get my luggage forwarded to my next flight (after customs/immigration) I would probably need to use a cloakroom – since I understand, I cannot check-in baggage for my domestic flight more than 2-3 hours prior to departure. I was hoping I could circumvent that!