1970's book where near-immortal people live in domes and a machine brings them back when they die

I believe it’s by a woman author. It’s her most famous book.

It depicted a world far in the future, where humans live in domes and are more or less immortal. Every time they die they return to a machine that tries to teach them and then return them to life in whatever form they prefer.

The narrator starts off as a woman and becomes a man later in the book and it deals somewhat with the trauma of that. She goes camping in a small dome and her dog gets killed. Wealthy people are fitted with wings.