Many thanks to those who replied. I see that I've been doing a lot of things wrong. You're absolutely right: eating dry food is not a natural thing for a parrot.
FWIW, until these two, I've always dealt with either songbirds (rehabbing and releasing) -- hundreds of them -- or precocial birds (ducks, chickens, geese) -- again, hundreds. I ABHOR cages. All the birds sharing my house had the 'flight" of the house. I/we simply draped towels over the furniture where needed. Washing machines are wonderful things to have.
When we first wed, my wife was on board with this. She bought birds like lovebirds and cockatiels, but had no problem with them flying around the house. But we lost a few -- one cockatiel escaped the house, a couple died by getting trapped behind our window blinds, and a couple of canaries drowned in our enclosed pool. I guess that's when they got confined to cages.
When we first got the two parrots, they had perches, not cages. And yes, they peacefully coexisted. The cockatoo had a manzanita tree to hang out on, and the Amazon a perch with bowls nearby.
They both were a lot more gregarious with us, too. The grey, especially, would walk down and across the floor, climb up onto my loveseat, and sit on my shoulder or hang out on the seat back. Except for the fact that we never seemed to have a decent TV controller (he ate the buttons off), no problem.
Ah! Now I remember: we got and raised a baby opossum, and he also had the run of the house. Couldn't have the possum and the birds commingling. That just wouldn't do!
The possum is gone now, but the birds are still confined to their cages. The biggest problem nowadays is, when they do get out, they set out to eat the furniture.
As far as socializing, Sterling will no longer climb up to sit with me. He just walks around on the floor. The Cockatoo, like all Cockatoos, loves to sit with me and be petted (he also wants me to bear his young <!>), but he too wants to chew on the furniture.
In 2001 we moved across the country from FL to AZ, driving with two dogs, the parrots & cockatiels, and the possum 2300 miles in an RV. In 2008, we did the same trip back. I guess the parrots haven't been quite the same since.
About the fighting: Yes, they did fight ... once. I had them both out on the floor, and the Cockatoo attacked the Grey. No harm done, because I was there to break it up, but we haven't let them out together since.
Trust me: My wife is terribly upset that these birds are confined 24/7. She loved having the birds all over the house, too. She now realizes that getting them at all was a Bad Idea. But it's kinda hard to undo what's been done.
Looking back up at all I've written, I can see that our problem isn't about food at all. It's about behavioral issues, caused by the fact that these birds are caged all day.
Now, if I could only fix the problem. Any suggestions? Short of replacing all the wooden furniture with gingerbread?
Oh, one other problem: Peaches doesn't like my wife (even though she's the one who raised him from a nestling.) Whenever he's out, he will chase her around the room, shouting "C'mon, C'mon." If he gets within range, he'll bite her on the foot.
Sterl, OTOH, is an equal-opportunity biter.
Would it be safe to assume that these two birds are more than a little big pissed off at us?