We sign an agreement that commits their non-refundable payment for a brief period (usually 5 days), during which time I call landlords, employers, and run background checks.

Our foreclosures provide you with a deed without a redemption period (unless it is a tax foreclosure) but make sure the foreclosing deed you are bidding on is in first position.

Being an agent in TN requires you to work under the supervision of a Broker for a period of time before you can get your Brokers license.

They advertise all over the Internet for 2 free workshops a day over a week period to see the "Guy".

Is it possible/legal to market this while still being in the due diligence period before closing on the sale?

With the right to renegotiate to longer period as long as both parties agree of course.

If the specific neighborhood isn't the issue and it's just timing then remember to make your future rental contracts expire outside of the slow period. 

@Alejandro Obando I wouldn't include the rent for your unit as income for the period you plan to occupy the unit, but I would include it for the time period you don't intend to occupy it. 

You could make an offer with a due diligence period to give you a chance to get inspections, and check for any environmental issues (cars leaking fuel, oil, etc... could be an issue). 

I am worried that transitioning PM companies within a short period of time would fatigue the tenants; however; neglecting a maintenance request would be frustrating for me if I was a tenant.