Hi pmw0813,
Thanks for posting, and congratulations on getting a new iPhone. As we understand it, you're unable to back up the old iPhone because you don't have enough space. Is that correct? We're happy to see how we can help out.
To double-check, are you making a backup using iCloud? If so, we recommend starting here: If you can't back up to iCloud
Check how much iCloud storage you have available, then check your Next Backup Size. Then take steps to reduce the amount of information you store in iCloud, or upgrade your iCloud+ plan.
- Go to Settings > [your name].
- Tap iCloud. The amount of iCloud storage that you've used is listed at the top. Learn more.
- Tap Manage Storage.
- Tap Backups, then tap the device you're using. Next Backup Size is listed under Last Backup and Backup Size.
It may take a few minutes for your device to calculate Next Backup Size.
- Make more space available in iCloud by reducing the size of your backup or deleting information that you don't need.
- Upgrade your iCloud+ plan.
- If you still need help, contact Apple Support."
It might also be worth checking out Quick Start to see if that would work for you: Use Quick Start to transfer data to a new iPhone, iPad, or iPod touch
Let us know if that helps. Take care.