Hi @MccorryIT,
Ideally, if the initial names of all the files in your specified folder do not contain any symbols, there may be ways to achieve your needs.
By judging whether the file name of the currently modified file contains "_", if it is not included, the current date can be directly added to the file name.
If it does, you need to replace the date with the current date.
If you have the specified folder, it is recommended to select the corresponding folder instead of Expression when configuring Folder Path.
Formula reference:
Condition:
@and(not(contains(triggerBody()?['DisplayName'], '_')),not(equals(triggerBody()?['IsFolder'], true)))
Create file:
replace(triggerBody()?['Path'],triggerBody()?['DisplayName'],'')
concat(first(split(triggerBody()?['Name'],'.')),'_',utcNow('dd-MM-yyyy'),'.',last(split(triggerBody()?['Name'],'.')))
original:
first(split(triggerBody()?['Name'],'_'))
extension:
last(split(triggerBody()?['Name'],'.'))
Create file 2:
replace(triggerBody()?['Path'],triggerBody()?['DisplayName'],'')
concat(outputs('original'),'_',utcNow('dd-MM-yyyy'),'.',outputs('extension'))
Image reference:

Note that the Delete file action is not configured in If no branch. This is also my confusion. If configured, all files will be deleted.
Edit:
This method still has problems. It seems that the Create file action in the IF no branch is similar to the Modified file. It is not creating a new file. I am improving.
Best Regards,
Barry