I bought 2 TB of new storage with Apple for my MacBook Air, but it says my disk is still full

No. iCloud Storage is for syncing and backups (it's not an external store space)

iCloud backups

With a Wi-Fi network connection, you can back up your device using iCloud. You don't need to plug your device into a computer to back up with iCloud. 

iCloud backups include nearly all data and settings stored on your device. iCloud backups don't include:

• Data that's already stored in iCloud, like Contacts, Calendars, Notes, iCloud Photos, iMessages, Voice Memos, text (SMS) and multimedia (MMS) messages, and Health data
• Data stored in other cloud services, like Gmail and Exchange mail
• Apple Mail data
• Apple Pay information and settings
• Face ID or Touch ID settings
• iCloud Music Library and App Store content

Learn how to back up your device using iCloud or how to manage iCloud storage and delete backups that you no longer need.

External SSD/Harddisk Storage?

FYI: iCloud Storage is meant for syncing and backup, it does not increase what you have on the Mac.

Yes, you can, but give it a test first by loading the files from external hard disk before you delete them from your Mac.