Should I be worried about my clearing process, or anything else, since my unit will be going to NTC and my packet is probably still at S-1?

This is something you specifically need to address with your leadership as there are too many different factors involved for anyone to give you an informed reply.

Talk to your unit leadership and/or the S1 personnel about your concerns. If you get nowhere, then discuss with your commander or the battalion commander using their open door policy.

Walk over to S-1 and stand there until someone comes to help you. It is really tough to give someone the "Poor Man's Brush Off" if you are standing right there.

Did you talk to your direct first line leadership about this? Squad leader? PSG? 1SG?

How long has it been with S1? You should be able to see it in IPPSA. There should be some type of Rear D too, even for HBL but should be some people on Rear D for NTC.

I'd be hitting up my leadership to go and get it moving if it's been sitting on BC's desk awhile.