For your first attempt, given the soil conditions you describe, consider growing the potatoes using a container or raised area method. It is getting a bit late to start the potato process here. Most garden potatoes we are seeing are up a foot already, probably nearing bloom.

For the long term, you can get good results by amending the native soil with compost and cover crops for 2 or 3 years. Potatoes will do fine in that kind of loose, deep soil (12-18" deep is enough). We have gotten 10's of pounds of potatoes in a small area (say 10-15 square feet).

Planting grocery store potatoes would be experimental, but if they show signs of sprouting, why not. We have used only seed potatoes intended for planting purposes.

Consistent watering is key for growing well-formed potatoes.

Ultimately, you just need to cover the spud with a couple inches of soil, then as it sprouts, add more. You can mulch with straw, but I would aim to at least have the plant in 6-8" of soil if you can swing it, then mulch above that for a little deeper hilling (about 12"-18"). I think mine have produced better when in contact with the soil vs just in the straw mulch (though others may not agree with that).

You could plant organic store potatoes if they're sprouting, but take a look around at places that sell seed potatoes, they may still have some, and at a discount and already trying to grow (at least that was the case last week in Olympia). Last year I planted some marked down seed potatoes from the Co-op and they produced, though it was of course later than what I had planted earlier in the spring. It is getting late, and your yield may be reduced, but I don't think it's too late if you get on it and plant them.