Your two sentences present some discrepancy.
Not all 1-dimensional local rings are semi-normal, so the first sentence has nothing to do with the second.
If all 1-dimensional local rings were Cohen-Macaulay, then all local rings would be Cohen-Macaulay by virtue of the definition of the Cohen-Macaulay property.
Semi-normalization can make things worse. Three lines meeting in a point that are not contained in a plane (say the 3 coordinate axes in 3-space) is semi-normal and it is not Gorenstein. On the other hand, 3 lines meeting in a point and contained in a plane is Gorenstein, but it is not semi-normal. Its semi-normalization is exactly the above union of 3 lines meeting in a point that are not contained in a plane.