I think you need to pay off the balance from your prior public housing before you will get accepted into new public housing, not just say you will pay it if you are accepted.
I just looked online at some reasons for denial for Federal & state public housing & Section 8 (but I don't know your state obviously)
For federal & state public housing, they will look at rent paying history (so I think they consider a balance owed as an issue).
It looks like they may (but are not required to ) reject your Section 8 application if you have been evicted in the past 5 years from a public housing program or owe money to any public housing program or ever terminated from the section 8 program (2 separate line items). So, paying off the balance would remove one of the negatives on your application