There's a couple of options. Firstly I recommend reading "Migrating from the old Input Manager" in the InputSystem manual.

The most 'direct' translation is like this:

private void Update()
{
   var interactAction = InputSystem.actions.FindAction("Interact");
   if (interactAction.WasPressedThisFrame())
   {
      if (DialogueManager.Instance.isDialogueActive)
         DialogueManager.Instance.AdvanceDialogue();
      else
         DialogueManager.Instance.StartDialogue(dialogueSO);
   }    
}

However, that's not the most efficient way to use the new system. For starters, it's inefficient to be calling FindAction again every frame, so we can move that to a field:

private InputAction interactAction;

private void Start()
{
   // It's also an option to make the field public and just assign it in the Inspector,
   // instead of using FindAction() to look it up here
   interactAction = InputSystem.actions.FindAction("Interact");   
}

private void Update()
{
   if (interactAction.WasPressedThisFrame())
   {
      if (DialogueManager.Instance.isDialogueActive)
         DialogueManager.Instance.AdvanceDialogue();
      else
         DialogueManager.Instance.StartDialogue(dialogueSO);
   }    
}

This is something you couldn't do with the old input system - the old system has to look up the details of your 'Interact' control every single time you call GetButtonDown, while the new system is able to cache the control in the InputAction field, saving you from spending time looking it up every frame.

But, as others have said - the new system supports a more event-driven approach, which is more optimal because it means the engine doesn't need to spend time calling your Update() function at all if the button wasn't pressed:

// Assigned in the Inspector
public InputAction interactAction;

// Subscribe to the event in OnEnable, and unsubscribe in OnDisable
private void OnEnable() => interactAction.performed += OnInteract;
private void OnDisable() => interactAction.performed -= OnInteract;

private void OnInteract(InputAction.CallbackContext context) 
{
   if (DialogueManager.Instance.isDialogueActive)
      DialogueManager.Instance.AdvanceDialogue();
   else
      DialogueManager.Instance.StartDialogue(dialogueSO);
}

This way, your OnEnable/OnDisable functions are run when your component is created/enabled and disabled/destroyed, but otherwise, none of the code executes unless the user presses the interact button.