I think others have given you good advice, and hopefully you can find a trainer/behaviorist you like if you choose to go that route.

I'd like to add something about when the dogs try to nose in on each other's petting time. I think it's unacceptable. If I am petting one dog and the other comes over and trys to nudge its way in it gets ignored. If they still persist they are put in a sit/down stay until I am done, and are then released with pets. I am alpha and I decide who gets my attention and when.

Oh and a tip for breaking up dogs fights (even reaching in for a collar can still put you at risk) reach from behind and link your hands together under the belly and pull apart by going straight backwards then zig zag from side to side as you go back. You zig zag to prevent the dog from trying to turn and get at you. This of course works when you have two people (one for each dog). Hopefully though you won't have to do that again.

Good luck!!