Moved WP site and now what about the database?

I set up:
OldDirectory.domain.com > linked to DB 1234
I then decided to Move the OldDirectory.domain.com to a new subdirectory

I created:
NewDirectory.domain.com > linked to DB 5678 (set-up by hosting provider)

I copied all of the files from OldDirectory to NewDirectory.
My NewDirectory.domain.com site works fine

However – this is my confusion
NewDirectory is still linked to DB 1234

I want to delete my WP installation and sub directory OldDirectory.domain.com

What steps do I need to take to link NewDirectory.domain.com to DB 5678, not db 1234?

AND make sure all of my links are connected to the DB 5678
so that I can delete OldDirectory and DB 1234.

My hunch, based on what I read, including documentation from my hosting provider is to do a
Backup of DB 1234 and Restore the data to DB 5678,
then redirect NewDirectory via config.php to db 5678

run some plug-in (not sure which one) to change all of the links in NewDIrectory WP files.
Then I would be safe to delete OldDIrectory & in turn , DB 1234 ??
Thank you –

??