Google App Script, updates to sheet are not being found by rest of script. How do I make it include updates previously done in the script?

I have a Google App Script which is supposed to run, collect information from a user, then update the spreadsheet that's currently open. After it updates the spreadsheet (edits some cells), I have more scripts that take the sheet and processes the cells according to the pre-existing code that processes them (no reason to copy the code over--it's already huge). However, what I'm finding is that the script is getting the old data when it pulls the cell's contents later. I can't figure out why.

From what I can tell (based on my work with Excel VBA), Google Sheets does NOT update the cells within the sheet (perhaps just with an active sheet that you're looking at) until a later point. So even though I run a method that sets the cell values, several lines later when the script (using a different function) requests the cell values for the same cells, it gets the old values, not the updated values. So it appears that Google just stores a list of changes to make and then runs them at a later time (perhaps when the script ends).

I cannot find a way around this, either. The script ends normally within 5 seconds and the updates are made visually at that point, but the error continues. I tried using Utilities.sleep(5000), and it doesn't change the behavior. The cell continues to show the previous data during the entire 5000 milliseconds, then finishes the script, then finally updates what appears in the cells, and the old data is still used. Is there a way to force Google Sheets to update what's on the cells and then continue?

Here's a snippet of my code (I doubt you'll be able to make it run).

//master is the name of the sheet.
  master.getRange(rowIndex,1,1,3).setValues([[newFirst,newPreferred,newLast]]);
  //New name designations
  
  Utilities.sleep(5000);

  // Run DirectoryUpdate
  directoryUpdate(Logs);//reads the rows of the sheet and changes people's names in Google and Canvas.  It is reading the old data.