LR keeps going to same album on startup

Some observations, these do not solve your issue. But you might have some issues downwind.

Lightroom Classic version: 13.0.1

V 13.1 is available. unfortunately no feature updates, just camera and lens updates and many bug fixes (non of them relate to this discussion). NOt sure why s few added cameras and a few added lenses rate an update, as opposed to what was clearly a bug fix. (should this have not been LrC v13.0.3??)

for bug fixes, see:

https://helpx.adobe.com/lightroom-classic/kb/fixed-issues.html

Graphics Processor Info:
DirectX: NVIDIA GeForce GTX 1660 (31.0.15.3623)
Init State: GPU for Image Processing supported by default
User Preference: Auto

GPU driver v536.23, not so bad, released in August, a newer one, v546.33 is available. but it does not solve any issue you may be having. But in general, for the Adobe products, keep those GPU drivers up to date. Adobe has a nasty habit of leveraging the latest drivers (mostly an issue at a large update or upgrade)

Library Path: E:\LR backups\2023-08-08 1710\Lightroom Catalog-2-2-v12\Lightroom Catalog-2-2-v12-v13.lrcat

That is of concern. That path implies you are either using a uncompressed backup, or you keep your original working catalog on the same hard drive as the backup. Very bad form. Backups should be on a separate hard drive. Hard drive breaks, both working and backup go poof

Assuming this relates to Lightroom Classic - "album" is terminology from the Lightroom cloud based system - the standard behaviour should be for the Filmstrip to show the most recently viewed folder, collection or smart collection (not the most recently created). Also to be viewing the same image as before, plus to have any highlighted selection of other images the same as before. If there is something else happening, re-setting the LrC Preferences may resolve this. 

https://www.lightroomqueen.com/how-do-i-reset-lightrooms-preferences/ 

There is also a LrC Preferences setting (General tab) which controls what is seen after each import of new photos into the Catalog. If checked, you will always then switch to view the Previous Import special smart collection (the photos that have just been imported). If not checked, you will continue to see whatever was showing before your import.

Please post your System Information as Lightroom Classic (LrC) reports it. In LrC click on Help, then System Info, then Copy. Paste that information into a reply. Please present all information from first line down to and including Plug-in Info. Info after Plug-in info can be cut out as that is just so much dead space to us non-Techs and it takes up vast amounts of scroll space making the reply less readable and less likely that others will bother with your post.

If you're talking about catalogs in Lightroom Classic, then that can be set in the applications preferences, first tab.

Otherwise, please clarify.