Need help backing up (due to camera hardware problem)

Hi all (apologies this is so long - it didn't look that long when I typed it...)

Am really hoping someone can help me out here. I bought a secondhand iPhone in November - it is still under warranty until next August so I can make a Genius appointment and should be able to get it replaced under warranty, but this phone has a lot of good memories and I get attached to stuff, so I'd like to try a few things before I get it replaced (plus, it still has its original front and back protective stickers on it!).

On Tuesday night, the camera just stopped working. I had been using it that afternoon without a problem. I was looking up something online on it and then turned it around to take a picture of something and it looked like it didn't detect any light - the screen was just dark. It seemed to freeze, so I pressed the Home button and tried it again. This time it went black again, but then the shutter closed. It now does this constantly. I've noticed that it has a split second when I can change things on the screen - I can swap it to video, but then after that it completely freezes the app. I can swap it to reverse camera - which also completely freezes, but when I close the app and reopen it, it actually works. Just not front facing. I've tried most things I've read online - playing with settings, gently shaking the phone, gently tapping near the camera. Nothing works. Sometimes when I think it has frozen, it has gone to a screenshot of itself and I can swipe sideways to the previous photo.

So, right now, my big problem is backing up my data. I know I have to back the stuff on it up before I make any changes. So I tried connecting it to my laptop (and it turns out that I don't seem to have ever connected it to my laptop before). It took forever for it to be detected, but it only gives me the option of setting it up as a new phone or using another backup. Obviously, if I do either, the files on my phone will be gone. All I really want to backup is the photos and texts. Notes would be good, but not essential. There were suggestions of just right clicking the iPhone and clicking Backup, but I only have the options of ejecting or restoring from backup.

I've seen a few programs around but I don't necessarily trust them. My previous phone was an Android and while I didn't like it, it had plenty of options on their app store to back up different parts of it. Do apps like this exist for Apple - I can't seem to find them - mainly photos and texts.

Would love it if someone could help me figure out how to back up my iPhone. Any advice on the camera would be great, but I can't do anything about that without making sure my files are backed up 🙂