Hello dakotashanice,
It sounds as though you're unable to download music to the Apple Watch. We've got some steps you could test out for this issue.
To start, are you an Apple Music or iTunes Match member? If you're an Apple Music member, try the following:
- If this is happening on a cellular model, test if this occurs when connected to Wi-Fi and only when connected to the cellular network.
- Try removing the affected content, then try adding it back again: Add music to Apple Watch
- You mentioned only the 'Recently Added' playlist is the only one you can select. Can you verify these other playlists are downloaded to the iPhone (if using Apple Music)? As these playlists will need to be downloaded on the companion iPhone, then synced to the Apple Watch.
Let us know how it goes, take care.