Can I transfer purchased songs to an older mac that no longer supports newest iTunes ( Music )?

Purchases from the iTunes Store since about 2010 have been free of DRM. You should be able to copy recent music downloads to any other computer, either over a network or by way on a portable drive. That said if you're running El Capitan through High Sierra you can install iTunes 12.8.2.3 and access your purchase history directly.

If you're using the same Apple ID then changing the password shouldn't affect access to past purchases. See Hide and unhide music, movies, TV shows, audiobooks, and books - Apple Support in case some older items have been hidden. Note the ability to redownload past purchases is dependent upon continued availability in the store. It is possible that some of the items you can no longer access have been removed by their rights holders. There isn't any solution to this unless you can locate an old backup or device that includes those tracks.

tt2

See Set up file sharing on Mac - Apple Support. Details may vary a little between OS versions but it should point you in the right direction. You can create a shared folder (e.g. your iTunes Media folder) on one computer and access it from the other and use that share to copy files.

tt2

Try enabling Network in the Finder sidebar, then you should be able to browse.

tt2

Try https://www.techrepublic.com/article/how-to-share-folders-to-your-network-from-macos-high-sierra/.

tt2

Try Finder > Preferences.

tt2

You're welcome. 🙂

tt2