I used to live in zone 5, near the ocean and had the most incredible soil; silt/loam. All of my plant beds are raised mounds no matter the type of soil so when it rains excess water drains out of the mounds down into my little trenches and away. The soil during the winter didn't freeze past a few inches so I left my potatoes in the ground all winter. Dinner time, I'd just go out and reach into the soil mound and pulled out perfect potatoes. Spring time we'd pretty much had depleted most of the potatoes. Those that were missed would start growing so easy to pull out during the growing season. You want to use clean, disease free seed potatoes every season and always rotate and plant in a new area of the garden.
This new place we live is very different. Last fall I hadn't harvested my potatoes when we had a surprise snow in early October that completely crushed our hoop greenhouse burying the plastic and all my potatoes/perennials under 3' of dense snow. It was impossible to harvest those potatoes and all I could hope for was that the snow would help insulate my potatoes. Well, every single one was total mush, sigh.
What zone do you live? Sounds like you didn't raise your plant beds and you've got clay or poor drainage. You do not want to wait too long and allow your soil to freeze. I'd get those potatoes out of wet, soggy soil as soon as possible. Don't worry about harvesting while the soil is wet. Take the potatoes out and lay out on newspaper in a dark, very cool place.
You will need to make a potato cellar, have you decided on a design? I am going to use a large insulated 'beer' cooler. Our land is flat and when it rains the water sits on top 6 -12" until it slowly drains into the soil. And our soil is like sand, it is volcanic pumice and you would think it would drain much faster. So my cooler will be buried below the soil leaving a good 6 -12" from the bottom of the lid above the surface of the original grade. Then I am going to cover it with breathable row cloth and at least a foot, maybe two of soil over all leaving access to open the lid. Look up potato cellars, there are many many ideas. You need a dark, cool, dry with consistent cool temperatures to store your potatoes, carrots, beets...root vegetables, onions, garlic and shallots. I am using wood pellets (the kind for pellet stoves) and newspaper to put my potatoes in within the cooler (s). Allow your potatoes to dry, do not wash, before storing.
Let us know what you've got going, how much land, does it slope, type of soil, do you have an unheated basement??? Your zone? Where do you get your seed potatoes? I have fallen in love with 'Potato Gardens' in Colorado. You can get garlic, shallots, and incredible potatoes. They are wonderful answering questions and the guy's son is this major soil expert. Did you know potatoes love an acid soil? I am so spoiled, I just don't want to eat another store potato...just bleh. I think after rereading your question this might be a bit more information than you need, oh well! Potatoes are the easiest, most reliable, rewarding crop to grow!!