No you can’t.
What you’re suggesting isn’t possible because of how AxCrypt works. The term ‘Key Sharing’ is a misnomer in the sense that you’re not sharing a key – you’re using another users public key to encrypt the shared file with.
As a user all you need to do is first select the option to share with your friend and second email (or upload to Dropbox) him the file. He only needs to use his password (and not yours!) and he can decrypt it.
Even if a hacker gained access to the file by breaking into your Dropbox he’d still be unable to view the file because it would be protected by encryption. Therefore you need two things: the file and the password; without both your files are secure.