Tenant Screening - Need advice!

[Vincent Plant]

I am in the process of leasing my first rental property. Its a low income property so of course there are a lot of people applying with "less than satisfying" applications and understand that as a land lord for this type of property I will have to overlook some things.  I have received several applications as of yet but I do like one. The credit is bad and there is a prior eviction but I have talked with her and she was very upfront with everything  and explained pretty thoroughly and it seems their situation is better now. 

My ultimate goal is to maybe market the property for another week to see if I can get something better, and if nothing comes through then see if I can work it out with this family. 

How would you suggest prolonging that situation? They paid $60 ($30 each) and applied and are expecting a decision asap. I want to wait it out and use them as a plan B if nothing else comes along in the next few days. What would be the best way to handle this without getting into deep water?