What's a better way of saying "I'm willing to sign a confidentiality agreement immediately" that conveys that my willingness to sign an agreement immediately is for their benefit but doesn't sound desperate, urgent, or like I'm demanding that they send me the agreement immediately?
Try at your convenience:
I can sign a confidentiality agreement at your convenience.
Here's one definition:
When one has the time to do something. No rush, you can fill out that paperwork at your convenience.
This phrase works because it puts considerations of timing to the other party. It focuses on their benefit rather than your own, while conveying that you'll make yourself available when it's convenient for them. It's polite and doesn't presume when is best for them.
This phrase is modifiable with earliest to signal that you'd be available as soon as they could get to it: at your earliest convenience. That said, readers will read this as more urgent, so I'd only use the additional adjective if you want to convey polite urgency.