I planted 5 of these in my side yard (full sun exposure) the weekend of 8/5 and 8/6. We have had moderate rain and I have used a sprinkler at least every other day as well. I noticed that 2 or 3 of the plants each have a branch where the leaves have turned a bright yellow. The remaining plants and branches have the green leaves with the golden outer edge with no change over the past two weeks. I researched japanese euonymus problems and this doesn't appear to any of the common issues such as scale.
It sounds like a root rot from all the water. This is not a water plant. Like most plants it does not like to sit in soggy soil for long. Plants like about 1" of water a week that drains and the rest of the time they like soil that remains about as moist as a wrung-out sponge. The top 1" or so of soil can completely dry out between waterings.
Since Japanese euonymus is a non-native invasive plant in Maryland (which means it escapes and damages natural and park areas), you might consider replanting with a different species of shrub. There is more about native and invasive plants on our website.