Would then e.g. Moridin as SH spy on Sammael and Graendal in ACoS, ch.40, or SH free Semirhage, though he is quite limited in this persona. Makes no sense.

We also see in ACoS, ch.25, that SH tortures Moghedien and then sends her to Moridin. This makes pretty clear that both can't be the same creature. They couldn't switch places and clothes that quickly.

Ultimately, I don't see the sense in fusing Moridin and SH together. It makes more sense that the DO could simply project his shadowy from into SH and this explains his powers/limitations. Moridin is simply the Forsaken, who is most dedicated to the DO and whom the DO trusts most, and therefore SH and Moridin are closely working together.

I would also think that Rand would notice a further direct link to SH, because of the balefire link to Moridin.