[Advice] Would I be "scrapping the bottom of the barrel" if I search 2 weeks before my move-in date?

budget: $2500 for rent and utilities
thing1: moving from out of state
thing2: move-in mid-September
thing3: will be in NYC for up to a week, without a broker

A lot of the guides I've read said that I'd be "scrapping the bottom of the barrel" when it comes to selections when I start too late, but I know it's high season, and I think my budget is pretty good too.

• Is it likely that I be able to find something in a week of searching without a broker?
• (for my particular situation) Will I be significantly limiting my choices by making the trip too late?