If I am on Apple Music, is there a way to block certain songs/albums from an artist, but still get recommendations for others in their discography?

Have you tried marking the songs that you like as Favorite or using the Love and Suggest Less options on the songs. Even though the song titles are the same, Apple Music should respect the album associated with the song that you have marked. These settings can be applied to Songs, Albums, or Artists.

For example with Taylor Swift, you can mark the original albums with the "Suggest Less" option, then mark some of her newer songs or albums with the "Love" option. If Apple Music continues to play original songs that you do not like, just use Siri and say "Hey Siri, I don't like this song". Apple Music will stop playing the song from that album, but should still play the current ones on a different album even if the song title is the same.