Answer: Based on your description, there is a severe drainage problem in your yard. I would strongly suggest you consult with a professionally trained and degreed landscape architect or other professional with extensive experience in correcting drainage issues and see what they suggest. Drainage is complex in that water always needs to run downhill, so it must be re-directed to a more desirable area where it can soak into the ground. Sometimes there is a fairly simple solution such as a swale and retention area, but other times it requires redirecting runoff collected over a larger area. This might be important in terms of protecting your home's foundation as well as making the yard more usable, so that is why I really suggest you get professional on-site advice.
For the lawn, begin by running some basic soil tests to check the fertility and pH levels. The test results will tell you if you need to fertilize and/or add lime to adjust the pH. That testing is the first step to getting a healthy lawn and it is the most important. Yellowing can be caused by many factors such as pest or disease, drought, lack of nitrogen, exposure to herbicide, overfertilizing, and so on. I would suggest you consult with your local county extension for help with diagnosing your lawn problem(s). Based on the test results and the symptoms, they should be able to help you develop a custom lawn care plan for your yard.
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