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.
I suppose you may mean "Are all 1-dimensional semi-normal rings Cohen-Macaulay?"
I think even that fails. If I am not mistaken, then 3 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 certainly not CM. Interesting to note that 3 lines meeting in a point and contained in a plane is CM, even Gorenstein, hypersurface, but it is not semi-normal. I believe its semi-normalization is exactly the above union of 3 lines meeting in a point that are not contained in a plane.