I need to transfer my text history and pictures from my old iPhone to my new iPhone, but have already moved all other data to my new phone using iTunes and spent time organizing it. How do I make that transfer without resetting everything?

For your pictures try importing them to your computer and then syncing them back to the SE.

Import photos and videos from your iPhone, iPad, or iPod touch - Apple Support

For your texts, did they move with the backup restore?

My understanding is that they should have. Are both phones on 9.3.1?

About backups in iCloud and iTunes - Apple Support

Then you can not send the texts from the old phone to the new phones without overwriting everything that already did on the new phone. Did you restore the correct backup? Or I should ask, did you backup the old phone and restore that backup to the new phone?

The text messages should have moved over to the new phone since you went from an iPhone backup to another iPhone. If you go from an iPad backup to an iPhone, they will not transfer.

I don't think this will work, but it will not hurt anything. Try a forced restart, hold down on the sleep and home buttons at the same time for about 10 seconds until the Apple logo appears on the screen.

The only other thing that I could suggest to try is to restore the backup again and see if it works, but that will overwrite any new content that you have added as you already know.