Being the neighborhood low spot is a tough challenge for you and for Ask an Expert.
The gray color and smell indicate that this has been a long-term issue, with water filling all or most of the air spaces in the soil. Healthy soil has about 25% air-filled spaces. Without this, the chemical and biological processes needing that air do not go on. The soil becomes anaerobic and most likely unable to support many of our garden plants.
Without the ability to get the water out of the soil, not much can change. The most impactful thing you could do is to install drainage features of some type—drainage tiles or lines or a French drain—that will move the accumulating water away from your garden area. This would probably include re-grading, as well. Once this is done, you could begin to build a more healthy soil.
Depending on the specifics of your spot, forming raised beds—making the plants’ root zone above the problem soil—could be effective. Mixing in organic material (compost, planting mix, etc.) and planting cover crops of legumes and grasses would get the beds going. Ensuring that these beds have good drainage would be critical. If standing water is allowed to invade these beds, you would be in the same position you are now.
The OSU Extension publication Raised Bed Gardening can assist you. Read or download it here
Regardless of which avenue you choose, you have an intense process ahead to change the existing terrain.
Consider gardening in containers. You can choose ones as large as horse troughs or ones just big enough for a tomato. These can be placed in the best growing locations or as ornamental features on deck or patio. The publication Growing Your Own has a good section on container growing.
Good luck with your 2017 garden.