How to make a spread sheet auto sort when updated with dates in two columns

I am trying to create a patient spreadsheet that has two columns with different types of dates/data In column D it has month due and column E has date patient is scheduled and I want it to sort the data by date when either column is updated. I also want Column D to keep the conditional formating so we know which patients need to be scheduled because they are past due (red) vs. the ones we have some time (green). Once a patient is scheduled the data in column D is deleted and column E has data entered into it. Is it possible to have the entire worksheet sort based on the column D and E data. I hope this makes sense, I have attached a sample worksheet because HIPAA laws, the real one has 83 patients.

You've presented a "Before" view with the attached workbook. Please provide in the same workbook (same sheet down below or on Sheet2)
how the data would look after sorting.

Reason: there are no correlations with the data in Cols D & E. And the names those dates are related to, do not repeat - they are all unique.

So data in Column D is when patients are due for their 6 month follow up appointment and Column E is the dates patients are scheduled for. I am trying to get it to sort the patient names when data from column D is deleted because we were abe to schedule the appointment with them so a date is then added to column E for that patient. So for example if you look at Sheet 2 if we Scheduled Sam Smith for 07/30/2021 I delete the May 2021 from Column D and enter the 07/30/21 date in Column E. I am looking to get it so the list will update so when we enter the actual appointment date it sorts the patient in the list of appointments scheduled and vice versa so after the patient had the appointment they will be due in 6 months so I will update Column D with January 2022 and delete the 07/30/2021 so I want to have Sam Smith then back in the Appointment due column on the bottom (see sheet 3) because he is not due for a while where the red are due sooner.