Seed Potatoes Problem

I bought some seed potatoes about a month ago. I didn't take them out of the clear plastic bag from the store. I put them out in my laundry room (bright shade) that is not heated like the rest of the house but not freezing. Now that I've prepared my beds I discovered that the potatoes have grown long white roots on many of the eyes. The potatoes don't appear to be moldy. They aren't mushy but not quite as firm as when first bought. The ones on the bottom seem to have been more moist.
What do you think, are they ok to plant or should I try and get new ones?

Roots? They're just doing what you want them to do. Seed potatoes do get a little... not squishy, but not firm. That's okay, they are fine.

You can cut up the seed potatoes, a couple eyes to each piece. I forget the rules there, but I think there's something about letting them dry after you cut them up. (Maybe someone with more experience can provide input here.) Or, just plant them whole.

I read in a potato seed catalog that seed potatoes should be cut (IF cut) 24 hours prior to planting.

I've done them both ways.
If you cut them be sure there are 3 eyes on each piece. Allow the cut side of the tater to dry indoors or outdoors in the shade so it skins over. That's when the cut side dries out a bit and has a smooth, harder surface. It should only take a few days.
Once its skinned over, you can plant it, eye-side up.

Cutting vs not cutting should depend on the size of the seed potatoes and whether or not you want to get the maximum number of plants out of your seed potatoes. If you have small seed potatoes, no more than 1.5" across, you don't need to cut at all, but if they have lots of eyes, you can cut the larger ones. Potatoes over 1.5" are often cut (but don't have to be) to get more plants for your money.

All that being said, with sprouts already growing, you will know exactly where to cut (2-3 sprouts per seed potato piece) and it supposedly matters less about letting the cuts dry if you already have sprouts growing (I've been told, haven't tried it myself). The "letting the cuts dry" part is more important if there are no sprouts present, yet, when the seed potato is planted. The seed potatoes can rot in the ground if the cuts aren't dried. It makes sense that with sprouts present, those sprouts are ready to take off (since they're already a plant) and it matters less if the seed potato rots, it's already done it's job of providing food for the next generation of plant growth. The seed potato pretty much rots away eventually, anyway, it only needs to last long enough to start that new plant.

Just think of it as getting a jumpstart on the growing season and plant away!