Hello pkBrooklyn.
Welcome to Apple Support Communities. I understand that when you connect your iPhone to your Mac that Photos open. You want to prevent this from happening when you get a new computer. I’m happy to help.
It’s possible that you may have selected the option to open Photos whenever you connect your iPhone. Please see the subsection titled “Tip” under the section titled “Import photos from a camera, iPhone, or iPad” here: Import photos from a camera or phone using Photos on Mac - Apple Support
I recommend restarting your Mac. Once your Mac has fully restarted, connect your iPhone to your Mac. Does the Photos app open? If so, are you able to de-select the option to always open when your phone is connected?
Are you able to back up your iPhone now? If not, what error are you receiving?
When you get your new computer, you have the option to not select for Photos to open every time you connect your device.
Let me know the results.
Take care.