My newest addition has very, very high food drive also. He is 3 1/2. He knew absolutely nothing but sit. I've had him a month now, I think,....I can't believe it's been that long already. But I was having trouble with him not paying any attention to me. He is just a giant love bug, but constantly mouthing and jumping and squirming and rolling when trying to teach him anything.
Obsessed with the treat, I was trying to marker train him, but he just wasn't getting it and I talked to my friend and agility trainer about him and I dug out the clicker she gave me a couple years ago I never used and had him on leash so he couldn't just leave.
The first thing I did was teach him to look at me. Treats covered in my hand, at my side. First time he finally glanced at my eyes, click/yes, treat. Took a little while, maybe longer than a little while to quit obsessing on the treat and finally calm down for a split second and look at me. I didn't want to use a clicker, I just wanted to verbally mark it. But my trainer thought it would be easy to phase out the clicker and use the marker because he just wasn't getting it with my voice. But that really helped us, you didn't say if you taught her to 'look' at you. I can't hold his eyes for very long yet, but it was a very big 1st step for us.

As far as biting your fingers when Maya takes the treat, put the treat in your palm and give it to her with an open hand. You're less likely to get nipped.