how does a family member access apple music under my family plan without signing up for a new account?

Assuming you have enabled Family Sharing and have an Apple Music Family subscription, they should see the ability to access Apple Music and an iCloud Music Library once signed in to the iTunes & App

Google is not always correct.

With an Apple Music Family subscription and Family Sharing in place, each member in effect has their own Apple Music subscription.

There is no "Listen to a family member's songs in Apple music"

You each have your own iCloud Music Library and playlists. Each member signs in to iTunes on their device using their unique Apple ID and sets up Apple Music.

Manage your Apple Music membership on your iPhone, iPad, iPod touch, Mac, PC, or Apple TV - Apple Support

Also see this...

If you don't see your family's shared content - Apple Support

This is the answer:

• You find the songs others in your Shared Family group have purchased by going to the iTunes Store. They're not visible anywhere in Apple Music.
• In the iTunes Store app, on the bottom row (Music/Movies/TV Shows/Search/More), choose "More."
• From the list that comes up (Tones/Genius/Purchased/Downloads), choose "Purchased."
• The next list should show you "My Purchases" and - if you're in Family Sharing - "Family Purchases." Choose the family member whose song(s) you want on your device.
• Choose "Music" (or, of course, "Movies" or "TV Shows," since those are also available).
• Either scroll down or do a search to get what you want.
• Once you've tapped the song, tap the down arrow in the cloud.

The fundamental misunderstanding is that family members' purchased music isn't automatically visible in your Music app. It has to be accessed via the iTunes Store. (I knew all a family member's songs wouldn't automatically be downloaded to my device. But my thought was that if I searched within Apple Music for songs not on my device, my family members' purchases would be visible.)

It's a mistake made easily enough that I suggest it be mentioned somewhere in a basic Family Sharing troubleshooting page.

I've been trying to figure this out for half an hour, and I've actually done it before. Family Sharing is enabled. We have an Apple Family subscription. I open my Music app, go to Library, see a list: Playlists/Artists/Albums, etc. . . . nothing that obviously takes me anywhere to family members' songs.

When I Google "Listen to a family member's songs in Apple music," all I get is instructions for setting up Family Sharing.

You're referring to iTunes purchased music.

The OP is asking about Apple Music.

Apologies for veering off topic. I don't subscribe to Apple Music and use the Music app the way I've always used the old iTunes app. Another innovation/distinction I have to learn to keep up with.

No problem. It's entirely possible the OP is confusing the 2 as well.

Nevermind....

GB