For government owned you would have to see the stipulations and hoops for taking this on.

If there's no lease or if the lease does not stipulate the terms of accommodation, the default in Texas for share living space is that the living room in a common area.

You could stipulate that anything under a certain amount of days they will be provided, and over is the responsibility of the tenant. 

Take no execuses/sob stories, if after moving in, they are late or can't pay, send eviction notices/start the eviction proceedings 3-5 business days (whatever terms/days stipulated in contract..check with your state).

That said, the actual money is not needed until an offer is accepted and then you have how ever many days your contract stipulates you have to get the EMD into escrow via certified funds (either cashiers check or wire transfer).For back-up offers, you will not have to put up any funds unless the first pending offer is cancelled and you jump from back-up position into buyer position.

Other than a minimum credit score what other specific stipulations can I legally make to help get the best tenant possible?

The only reason not to let them go now is I assume they have a stipulation on the contract saying they get paid on any lease with any tenant they’ve placed.

There may be a stipulation that allows payment within 24 hours with a deposit...still trying to find out if that is applicable in our area. 

Even if their trust funds run out, the law still stipulate certain amount to be paid out to qualified seniors of the society.

If not get it under contract with the stipulation that you can verify payment history.He has to have a bank account with deposits.