I'm in zone 5, bordering on 6 and warm-weather crops like tomatoes and peppers have to wait until May, sometimes LATE May, to be planted. Our late winters are usually very wet also. But don't let this stop you from planting cool-weather crops nice and early. If I had to wait for the garden to warm up and dry out so that I could run the tiller, I would be waiting a long time.
I went out the other day, March 3rd, and planted two types of lettuce, spinach and Sugar Snap peas. If all goes at least halfway decently with the weather, I should be eating salad by the end of April or the firt week of May. Many people wait until they can run their tillers to start planting, but here is what I do.
Take a grass rake and clear the leaves and sticks from an area you want to plant. Then scrape the ground (it was pure wet muck when I did this) with the rake and loosen up the soil. You don't have to go very deep, I probably only loosened to about 1 inch deep, if that. Then scatter your seeds in a wide row, my rows were about 1 foot wide. While holding the rake straight up and down, LIGHTLY tamp down the area with the flat side of the rake.
I didn't have any dry, loose soil to cover it but tamping the seeds down in that rough soil should suffice. Just for a light mulch, I grabbed handfuls of some grass cuttings that were in a pile from the fall, and lightly covered the seeded area to give it a light mulch.
I have been doing this for some years now and it usually works well. I have even had snow on my spinach before, but it didn't harm it. I am a very competitive person and it's always fun to eat a salad from your garden at the beginning of May, but I make sure to tell my gardening friends that I did so! Yes, I like to rub it in. This is about the same time most people are thinking of buying their seeds, but you will be eating from your garden already.
Here is a picture of what the garden looks like right now. Yes, it is trashed and I have my work cut out for me. But I still was able to plant the other day on the back side of the garden where it is cleared somewhat.
This will be a fun picture to look back on later on this year, after I have everything cleared out and the veggies planted.