To be honest I don't think humans are likely to go extinct from climate change any time soon. We're at one of the coldest points in Earth's history right now, with a global average of ~60f / ~15c -- but even at the warmest points in history, where the global average was something like ~80f / ~27c and most of the earth would have been completely uninhabitable for humans, the poles didn't get above ~70f / ~20c in summers.
Society and industry as we know it would likely collapse by the time we hit a ~7f / ~4c increase in global average, so that gives at minimum a ~13f / ~8c runway without major human interference for the climate to steady itself, which it's done in the past. Of course, at that point we'd probably be reduced to scattered tribes across Greenland and Antarctica. Who knows if an industrial revolution would ever happen again. But there would be enough land and resources for subsistence living.
For better or for worse I think humans will probably be around until whatever tipping point where complex life as a whole gets killed off. Which kinda sucks, since we fill niches that other, potentially less terrible species could evolve to fill in the future. Humans only came around because the dinosaurs (mostly) died out after all, who knows what neat species could come after us if we made room.