Unfortunately, iCloud storage isn't going to help in your case because iCloud is just online and lets you see photos that are on another device or computer to share with all devices.
In order to free up space you'll want to have a PC or Mac with iCloud Photos. Transfer all your photos to the Mac or PC, this will let you put them in iCloud and you will be able to still see them on the iPhone. After saving the photos to the PC, you'll want to erase most or all the photos from the PHONES memory, which will free it up to take new pictures.
Your pictures will be in iCloud, just save on the computer, and you will be able to SEE them on the iPhone, but won't be saved to the iPhone.