Iīm happy things are getting better for you. When I first read your first posting I was surprized because it was something I could have written 11 years ago!
Itīs sad some things doesnīt change. But at the same time Iīm glad to see improvement of the thinking of the people that write answers. Of course not the same people
but still that shows me we are going in the right direction!
I canīt share my experience how I resolved my problem because of my 11 years of training I have moved around and changed around 10 dojos including one dojo of my own.
I have seen that moving to a new dojo is like starting a new job or something else where you have to make a new impression. I had to show everybody including the sensei how I should be treated.
In different dojos there are different ways of choosing uke. Some teachers use everybody once, others everybody from the advanced once, others use their favorites and other unfortunately only male favorites. You can not change a person and the same goes to teachers. You can not change their way of teaching and choosing ukes. But of course you can increase the times being uke by learning more about ukemi and all the other things people mentioned as advise in this thread.
I started trining in Sweden and there you have white belt until you get your hakama at 3 kyu and then black belt and hakama. On my first seminars I cloud observe how after the sensei showing the technique people fast choose partner and being a female white belt I was last to be chosen. Of course the black belts goes first, then the hakamas then the males then the females of the whites. Of course when I got my hakama I was not the last to be chosen and so on... Itīs so sad
Many thanks for the inspiring replies in this thread!