Why is my iPad showing full storage despite having available space?

Assuming that your iPad has at least 4-6GB of available free-space, the message that you see concerning insufficient storage perhaps originates from your Mail Provider.

Many Mail Providers limit the maximum storage available on their Mail servers for their customers email accounts. These limitations are common, in particular with "free" email accounts. If this applies to your situation, you will need to delete Mail messages from your Mail account before you will be able to send or receive email.

What exactly did the message say that it was full?

Also, perhaps this article will help: If you can't receive email on your iPhone or iPad