How to retrieve photos from a broken iPhone?

So basically my iPhone 6 cracked and then got slightly water damaged. There were problems before with the phone in which it wouldn't let me back it up to the cloud. Now the phone screen turns black when you click on an app (like settings.) Even though the screen turns black, the app is still open. I just got a new iPhone X and realized that the last back-up was in March. I'm missing a lot of texts, important photos, and apps. I was wondering if there was a way I could retrieve the lost things and put them on my new phone (preferably without a computer) I still have the 6. Could I take the SIM card out of the 6 and put it in the X just long enough to back up everything? I don't really want to go through the trouble of getting my old phone fixed.

Photos aren't stored in a SIM card. If they aren't in the cloud, you will need a repair shop.

Your SIM doesn't hold anything of use to you, moving it to another phone will just allow you to receive calls and texts to your number.

You won't really be able to do anything without a computer. Your best option is get a computer and backup using iTunes.

Alternatively you can plug it into the computer and install QuickTime, from there you can view your iPhone screen on your computer by going to File > New Recording and selecting your phone in the drop down box. You can use this to try and navigate through your phone settings to send yourself the photos or back them up.