I think a new normal, in the sense of a new steady (less favorable) state of the climate, is not something that is being suggested by the science. Just to point out one relevant fact, consider that the current CO2 levels were present at least going as far back to Pliocene geological epoch https://en.wikipedia.org/wiki/Pliocene_climate. I think it is reasonable to expect at least plenty of instability as the Earth gets closer to the temperature and sea level of that time.