Q: how do I download my iPhone to computer so that I can redownload on new iPhone?

The following has instructions on how to transfer to a new iPhone via either iCloud or iTunes on a computer: Transfer content from an iPhone, iPad, or iPod touch to a new device - Apple Support

To transfer data from iPhone to your new phone, you can use iTunes or iCloud to backup your old iPhone and restore to your new phone.

Transfer content from an iPhone, iPad, or iPod touch to a new device.

If you think it's a little time-consuming and don't want to restore from a backup,

you can try some third-party tool, like MobileTrans, a phone to phone transfer tool.

BTW, if you just want to transfer the texts in Camera on old iPhone, you can mount old iPhone as an external hard drive,

find the DCIM folder and export pictures to computer. Then, use iTunes to sync these pictures to your new phone.