The further North you go, the greater the difference becomes when the climate changes. The temperature at the Earth's equator only dropped 2 degrees during the last ice age - meanwhile, Minnesota was covered under huge glaciers. The temperature in Puget sound wouldn't be that different if all the Earth's ice sheets melted. The Oceans tend to prevent temperature swings from going too far one way to the other - the further from the Ocean you go, the more extreme the temperature swings in both directions become. It might rain even more there than it already does in Puget sound. The warmer the Earth gets, the more water evaporates out of the oceans, and all that rain has to fall somewhere.

Prediction is always an extrapolation but the question is how many issues do you extrapolate, in reality, everything is affected over time but the assumptions of seawater rise is based only on the amount of land ice being melted ( the north pole is sea ice which would not raise sea levels but Antartica and Greenland are land ice ). However, there are other possible factors such as the spin of the Earth, water flows so it would tend to flow towards the equator and it's added mass there would move more gravity in that direction pulling more water, it's been suggested that water levels might even recede closer to the poles but it's likely it won't rise as much as most expect closer to the poles but would rise more than expected towards the equator. Also, sealevels is just one possible effect, there are others such as ocean salinity, the salt level in the oceans as ice is fresh water. The salinity affects ocean circulation in the Thermohaline circulation which distributes the heat from equatorial regions towards the poles and brings the oxygenated cold water and nourishments back ( phytoplankton like the polar regions and cold water holds more oxygen ). If the circulation stops, the regions closer to the poles such as western Europe might cool down. We really don't know what will happen because we can't think of every possible factor, the ones we can come up with are in various computer models but every year, someone comes up with another factor that was overlooked.

The conditions which could melt all of the ice sheets would make most or all of Earth warmer than it is right now. Much of the land in the Puget Sound area will be under water.

Man haven't you seen the day after tomorrow DUDE! WE'RE ALL GOING TO DIE MAN! IT'S ALL A CONSPIRACY! SAVE YOUR PLANET BEFORE IT'S TO LATE MAN!

The ocean levels would rise, and there would be a bigger chance of flooding in your area. But other than that very minimal change.

I know, it's highly unlikely that they'd ever do so, but out of curiosity, how different would the Puget Sound region feel, would it be more humid than it already is, or would it become more arid?