The ISWC (International Standard Musical Work Code) database is a good source for looking up which songs have been registered to a songwriter, and in what order.
All you need is the IP of the artist, that you get for example from Songview
The IP for Lana / Elizabeth Grant is 00579892274
They show no registration date, but you can sort the results by rank or preferred ISWC (some songs have archived ISWCs, so they may have been registered earlier).
I find the registration data very interesting, including which songs have been registered recently but not released.
P.S. unfortunately, this search does not seem to be complete. For example, the song American is missing here, although it has an entry in the database. So it is recommended to do a parallel search in Songview.