Greetings and salutations STYT13!
From what you've posted, it seems that the Music App on your iPhone isn't reading your music tracks correctly since you updated to iOS 9.3.2. When I play my music on my iPhone, I expect it to play everything as well. I'll make sure you can hear the entire song.
Before beginning any troubleshooting, I always recommend you back up your device and import the pictures to make sure your personal information is safe.
How to back up your iPhone, iPad, and iPod touch
Import photos and videos from your iPhone, iPad, or iPod touch
Erasing and resyncing the music on your iPhone is a great first step. The quickest way to erase the music on your phone is by going to Settings > General > Storage & iCloud Storage. The tap "Manage Storage" under Storage and not iCloud. Tap on Music > Edit to select the red minus sign then delete the music only. After your music has been removed, you can sync your music back to your iPhone.
Sync your iPhone, iPad, or iPod touch with iTunes on your computer using USB
The next thing to try is to reset all settings on your iPhone. Settings > General > Reset > Reset All Settings.
What Reset All Settings does: Reloads all default settings: forgets your Wi-Fi networks, removes changes to Alert Styles in Notifications, Turns off Locations Services for third-party apps, sets Sounds, Brightness, and Wallpaper back to original configurations, removes Battery Usage statistics, turns off Siri, turns off Cellular Data, disables Wi-Fi sync, sets Auto-lock back to 1 minute, removes added International keyboards, and turns off all Accessibility options.
What it doesn't do: This option does NOT delete anything - pictures, contacts, music, apps, nothing. All media and contents will remain after the phone resets.
If the Music App is still passing over song intros, the problem could be how the iOS was installed. Since the problem you've had may have taken time to reproduce by the time you get to this point, you might want to back up and import photos and videos from your iPhone again. This procedure will erase everything from your phone.
Use iTunes on your Mac or PC to restore your iPhone, iPad, or iPod to factory settings
The first time through, set up your phone "As a new iPhone" instead of restoring your backup. Log into iTunes and iCloud and sync your music and information to your phone instead of restoring the backup at this time. Once you're confidant that your Music App is playing the whole song again, you can restore your backup.
Restore your iPhone, iPad, or iPod touch from a backup
Happy listening!