Plex should still search for lyrics even if you provide your own. It will default to your local ones but the ones plex searches for should still be available. Do you not see both sets of lyrics?
When playing music in Plexamp, I only see the local lyrics.
I tested it with a song that had synced lyrics that Plex provided, then I added a .txt file to the folder with random text in it, and Plexamp only displays this random text now.
I find the same behavior you did and MovieFan said, it defaults to local lyrics.
And If you add both types of local lyrics, it prefers the synched ones.
Elan provides the answer about switching below.