So I drafted one of these guys and I'm wondering if I've met his demands already and he's considering it, would exceeding his demands (unsolicited) push the needle in anyway? And if so, by how much?

I don't believe prodding them does anything. You'll just get another reply that he's still considering it, you have to wait (usually a week real time) till he accepts or counters. I've had players accept my initial offer, I've had them ask for an increased offer then sign right away... one time I offered above a player's ask just to see what it would do and his agent came back asking for less than what I'd offered.

Don't offer more.  Wait for his response. 

Haha sounds good, thanks for the help