How wet is too wet? If there is standing water, it's too wet. Otherwise, much depends on soil type. If you grab a handful of soil and squeeze, does it stay in a firm lump after you let go and poke it a little? If so, it's too wet to till...
If it breaks up, you can till and plant. The higher the clay content, the bigger the problem with excess moisture.
This might be a good year to make some EarthBox clones... they are designed to drain the soil quickly, and maintain adequate (but not excessive) moisture levels. Google the phrase, and you'll find various plans for making your own. If you can, use sturdy Rubbermaid type containers... the cheaper boxes will work for a year or two, but break down faster, especially in full sun (I have some that are 10 years old in my greenhouse which are still fine... the greenhouse blocks a lot if the UV rays.
You also might want to consider making "raised rows" in the garden... they'll drain quicker and allow roots to breathe.
My raised beds are doing well... herbs, radishes, greens, carrots snd beets. The peas are all sprouting well, and we planted a couple bushels of potatoes yesterday. Our official last frost date has always been (traditional... nit this stupid "last Monday"l Memorial day, so while in recent years we've been able to plant earlier, it's really not too late yet. But the soil is a lot cooler than we want it for warm weather crops.
I'll be pre-sprouting all the vine crops today, and am hoping to plant them next week. Sweet coen and beans need to be planted soon, as well.
Summerthyme