An update for everyone who wanted some more info on this: I went to get my US passport today at 7am at the NY passport agency. By 10.30am I had received it- with a bit of pleading and begging. My appt with Travisa was for tomorrow (didn't think I'd get the passport in one day) but I called them and they said the appt time didn't matter, to just come along. So I went by 11.30, but the security outside told me those at the call center don't know a thing and say whatever they want. They said the line was too long right then but to come back at 3pm when it was walk-in. So I went back home, went online, actually found an appt for 3.20pm, booked it and went back at 2.30pm.
A different security guy told me that I should go for a walk and come back when it's my time. Hardly anyone was in line at that point and I told him it was an emergency and besides, wasn't it 'walk-in' from 3pm anyway? He said, 'oh yeah' and then let me in.
Submitted my paperwork and was told it will take 3-5 working days so keeping fingers crossed.
It seems from this experience that no one in Travisa really knows what the rules are. Different people say different things and it was a bit confusing. I specifically heard one security guy tell one of the people in line not to pay attention to the 'rules' listed online (in relation to the OCI I think but I could be wrong) because they were incorrect.