...but now seems a good time to start turning over the ground.
Eeeeeeeeeeeeeeeeek! No, it isn't. All you'll do is compact the soil - it's way, way too early, and the soil is way too wet to be even
thinking about touching for at least another month. And apart from the leafy greens you mentioned, the planting season for tender crops such as peppers, corn, and basil is when the soil is actually warm - late May may be all right. Note that I said "may"....
You can start seeds indoors in April if you like for the peppers and the basil, but don't screw around with the corn. It doesn't transplant well at all, and if you put it in too early, the seed will rot in the ground. Trust me on this, my friend. Wait, and your patience will be rewarded.