You can't add memory to an iPhone. What you bought is more iCloud Storage, which cannot and does not add memory to your phone. It ONLY lets you store more backup data in the cloud.
You need to update your iPhone to iOS 16 (current version is 16.4.1) because your watch is clearly on WatchOS 9, which can only be paired with an iPhone running iOS 16 at a minimum.
If you are out of space on your iPhone, the only way to make more space is to delete things you don't need, such as photos, apps, music, files, email, messages etc.
One of the biggest users of memory on iPhone is photos. You could transfer your photos your computer for safe keeping, then delete them off your phone as an example. See this --> Transfer photos and videos from your iPhone or iPad to your Mac or PC - Apple Support (SG)
When you buy your next iPhone, buy one with more memory. You clearly don't have one with enough memory to meet your needs now.