@Lallu Tappu. "A beautiful family of 2 adults and 3 kids recently liked my house and are willing to rent it out. They told me upfront that they had poor credit due to medical bills and the...". This makes you sound desparate. Are you willing to lend them your asset?
Where do these prospective tenants fit within your criteria? As a result of your due diligence, Would you lend them money/cash taken from your pockets on a promise they would pay you back over the long term? What proof have they showed you that they are already taking care of those outstanding bills? Is it reflected in their current credit reports or is it something they just 'started' paying back? I believe in helping those getting back on their feet and heading in the right direction, but I don't want to be the test case if they are heading down the right path. i prefer my tenants to already be on that path so I have a general preference of a 5 year look back for common risks like credit and housing history.
I don't always agree that charging a higher security deposit is always the answer. In this case, I would require a perfect co-signer. I'd make prospective tenant show me that someone already believes in them in their current situation. Anyone can agree to a higher security deposit but it harder to find someone to back up any financial irresponsibility.
Don't change your criteria to justify accepting a poor tenant. Be patient and you will be rewarded with a great tenant.