A budget could very well be blown by hidden mold, asbestos, termites, leaky water, etc.  

I suggest you invest in an attorney reviewed and edited contract, draft the contract first if you want to tailor everything to your side, saves you money also, imagine all the worst case scenarios, over budget, over schedule, change orders, etc. you need to get the CO clause at a minimum pay, say actual hours x fixed rate and major material receipt only plus 15%.

Thats an extremely tight budget, as you know.

Always remember to do the math analysis and try to look at your prospective properties before you make an offer and possibly back out of the deal if something is found that is major and you did not budget for.  

The last time HUD cut the SEC 8 budgets - by 8% across the board I believe - every housing office decided how to apply the cut.  

Transactional funding would be my only option as I would not have the money to purchase from the seller outright with my own cash (on a really tight budget with this).

Did you not budget in these items and lined up the money before you signed the contract?

Can anyone recommend a contractor in the Northern New Jersey (Bergen County) area that they trust, and would be willing to walk through a property with my partners and I to provide a realistic rehab budget?

Areas to avoid  depends on your price ranges and family situation and especially your Budget .  

I also have suddenly hit the busiest stretch at my day job in half a decade, and I am over budget on the unfit already.