Hey Cheese,
I am so glad to hear that you have your important photos backed up safely. To answer your question about albums that are not shared and whether they are stored online and are easily available, I would like you to look at this article - iCloud: iCloud Photo Library overview
Basically if you use the iCloud Photo Library, all of your photos from the devices using it are stored online and available either to those devices or by using the iCloud.com web site. You can reduce the size of the photo files that are stored on your device.
To access the iCloud Photo Library from your iPhone use the information in this article - iCloud: Use iCloud Photo Library on your iOS device
For your Mac, use this article - Photos for OS X: Access your photos and videos everywhere with iCloud Photo Library
So using the iCloud Photo Library can, used properly, save some of the space on your iPhone. See this article - iCloud Photo Library FAQ.
Specifically -
iCloud Photo Library automatically keeps all your photos and videos in the original, high-resolution version. Follow these steps to choose how you store your photos and videos on your device:
- In iOS, tap Settings > iCloud > Photos or Settings > Photos & Camera, then select a storage setting.
- In OS X, click Photos > Preferences > iCloud, then select a storage setting.
If you turn on Optimize [device] Storage, iCloud Photo Library will automatically manage the size of your library on your device, so you can make the most of your device's storage and access more photos than ever. All of your original, full-resolution photos and videos are stored in iCloud while device-size versions are kept on your device. You can download the original photos and videos over Wi-Fi or cellular when you need them.
If you turn on Download Originals, iCloud Photo Library will keep your original, full-resolution photos and videos in iCloud and on your device. Download Originals is the default setting for iOS devices with the free 5 GB storage plan and for all Mac devices.
Thanks for using Apple Support Communities.
Happy computing.