This is something I've thought about quite a bit too. I like the Ozarks where I'm at a lot, it has a lot of good features, but I am worried about how hot the summers will get in the future and increases in droughts and flooding in this place that's already a land of extremes in many ways.
I consider climate change a very real and serious issue, but I'm also skeptical of specific predictions regarding its effects because of how complex the climate system really is. However based on what's been happening so far and paleoclimate evidence of the far past during times when CO2 levels were higher, I think we can expect significantly higher sea levels (although much of that rise will likely happen even farther into the future than 2100), warming most pronounced at higher latitudes, overall warmer averages most everywhere (but erratic cold snaps as well in many places because of jet stream changes). The changes in rainfall patterns are even trickier than temperatures to try to forecast. Warmer overall means an intensified water cycle, and more average precipitation overall. However that extra precipitation will likely be very unevenly distributed both geographically and temporally, much of it falling in heavy downpour events, and increasing temperatures also means increasing evaporation rates and increasing chances of flash droughts.
Looking at the lower 48 states from a purely climate change perspective, I'd put my bets on the higher parts of Appalachia, the interior northeast, and the upper great lakes region, particularly around lake Superior. The reasons are summers that are cool enough currently that they could get a decent amount hotter without becoming too extreme, and reasonably high amounts of annual precipitation that is also well distributed through the year, including the summer months, and moisture from multiple sources. Of course there are other issues going on than just climate change, and everyone's situation is unique so some other area might be best for you.
The Olympic peninsula is probably a decent choice as well, possibly one of the best spots in the west if above sea level a bit. The issue I see with the Pacific northwest in general is with the lack of summer rain, the summer dry season likely getting longer, hotter and more intense. I'm wary of anywhere that depends highly on winter mountain snowpacks, as warmer winters mean that even if precipitation stays the same or even increases, there will be less snow in all but the highest elevations because more of it will be rain. Being near the North Cascades might actually be better than the Olympic Peninsula, as they are higher elevation and the snowpack will thus be resilient in the face of more warming. Oregon will be worse off than Washington, as the Oregon Cascades are mostly not that high except for a few volcanic peaks. I expect the Pacific Northwest climate to become increasingly more like California's climate is now. The issue is that the ecosystems currently there aren't adapted to that extent of heat and drought and there will probably be mass tree die-offs and huge fires there when California-type dry seasons start happening, although this could be mitigated in local areas by good management.
Areas of the west that get the summer monsoon such as Colorado are more of a wild card, some say the monsoon could get more intense, and the mountains there are high and cold enough the snowpack declines probably won't be as dramatic as many other western states. However, Colorado is pretty dry to start out with except in some really high-elevation, cold areas, so it wouldn't take too much to push it over the edge.
The Appalachians/interior northeast/ upper great lakes, on the other hand, all have relatively high summer precipitation. I do expect droughts to worsen a certain amount in those areas, but IMO they have more likelihood of staying moist enough that the ecosystems will transition more by warmer-climate-adapted species increasing and cooler climate species declining, rather than by massive fires and die-offs of whole ecosystems. I have spent some time in southern Appalachia, and it's amazing just how many different sorts of weather patterns can bring precipitaion to those mountains. Unlike the Northwest, where the wet on the west side/dry on the east is very dramatic, the Appalachians have moisture coming from many sources. Moisture can come from the west, from the south, and from the Atlantic to the east, even from the northwest behind a cold front that's dry in most areas but then hits the mountain ridges that run perpendicular the wind and squeezes out the moisture. Who knows what unpredictable effects climate change will have on rain patterns, but areas with many sources of moisture are more likely to still end up getting enough of it.
Another thing to consider is that looking at a map of the world, there are very few dry climates along and near east coasts of continents (presuming they're facing an actual ocean, not a smaller body of water such as the Red Sea). There are many deserts and semi-arid areas along west coasts of continents, by contrast, as well as some very wet climates. I suspect climate change will push deserts on west coasts northward into some areas that are now Mediterranean climates (southward in the southern hemisphere). Eastern North America is unlikely to dry out as dramatically, even if flash droughts get more common and the heat and humidity in many lowland areas gets more and more extreme.
I haven't mentioned Alaska. I do wonder if at some point parts of Alaska will become desirable places to live. I've never been there and don't know enough specifics to have a strong opinion, but I imagine the more extreme rate of temperature change there will lead to convulsions in the ecosystem, and things like melting permafrost will wreak havoc in the places that have it.