Hi @Alyda !
I'm in the Mid-Atlantic region, near the Chesapeake Bay. I don't think we get quite the same amount of rain that you do in Seattle, but we get plenty, with high humidity, and lots of wetland areas. My personal recommendation is waterproof shoes, with wool socks.
No shoes are truly 100% waterproof in all conditions. No matter how hard you try, your feet may still get wet, or at least damp. Water, including dew on long grass, can run in around the ankle, and, if your shoe isn't breathable, sweat will develop and linger, That being said, a good, well-fitted WP shoe, with wool socks, will keep your feet as warm, dry and comfortable as is possible.
Another recommendation, if it's particularly wet where and when you hike, would be to add a pair of gaiters.
Waterproof (ie gortex and similar) shoes and boots are quite effective at least for a couple of seasons and if they don't get punctured. They definitely are a lot less work than leather footwear which you have to careful to treat with special waxes to keep them water resistant.
Just bear in mind that waterproof footwear takes much longer to dry out if they get wet on the inside.
The choice between shoes or boots will depending on where you plan to hike, how deep the water can be, how long you plan to hike and how much protection/support you prefer.
If you are not hiking far and it is really raining and or muddy then good fitting "rubber" aka "Wellington" boots work well.
Probably just as important is the choice of socks...you should use wool blend or synthetic to keep your feet warm when wet. Do not wear cotton socks. I personally recommend the "Darn Tough" brand wool blend socks.