If I am the landlord and if decide to accept the late rent due how do I stop the process after the order of eviction was filed?

First, I would not withdraw the Order for Eviction until the rent is actually paid. If you withdraw the order based on just a promise to pay, and then never receive the payment, you would have to file a motion to get the Eviction re-entered (at the least) or may even have to start the whole process over. Second, you can contact the bailiff and indicate you no longer want the order enforced. If it is in 36th District court, you may have to go to the court to find out which bailiff has the order. Expect to pay the Bailiff something if they have already acted on it. That amount should also be the responsibility of the tenant. Third--get the past due rent paid before doing anything. Yes, I did that twice for a reason.

Verbal promises to pay are worth the paper they are written on. If they haven't actually given you money, I would do nothing different.

Tell tenant as soon as you are paid in full you will call off the eviction. You can call the sheriff and have it delayed but not cancelled (Just because you get an order for eviction does not automatically mean they will be evicted.. I thought generally the Sheriffs in 36th district court will ask for $$ up front before they will initiate an eviction.