Speaking from experience, when we were children, our dog had to be "given away". He was given to a nice farmer with loads of rabbits that he could chase all day.
Of course, many years later we know that this was not true, he in fact had to be put down. I remember crying when they were taking him away, but do remember telling everyone after that he was having great time playing with rabbits in his new home. No long term psychological damage done ..............! My parents did the right thing and I think you did too.