Hello OliverCovey,
Thank you for using Apple Support Communities. Based on your post, when listening to Apple Music it skips every other song and skips through new playlists. We'll do our best to assist with this.
You've completed some nice troubleshooting here. Reinstalling the Music app should have resolved it. Since it hasn't then we'd try the following:
- Test this isn't related to Bluetooth. Go to Settings > Bluetooth and disable it. Does the issue persist when you test it out?
- Try another Wi-Fi network. Does this persist on another network or when connected only to cellular?
- Have you made any changes to your Apple ID region? As not all songs/content is available for all regions.
- If using Apple Music on other devices, do you have Sync Library enabled? See: Turn on Sync Library with Apple Music
Should it continue to occur, please reply back including the below details for more help:
- What changes (if any) were made to the device or your Apple ID two months ago?
- Did you change Wi-Fi providers?
- Are these songs being streamed over the network or are they downloaded to the iPhone?
Wishing you the best, take care!