It's not explicitly said the IQ will always be constantly plagued by the voices, but neither does it seem to be something she can just switch on and off at will. If you've made Morrigan drink in your playthroughs before, think of the scene at the war table where she's concentrating to make sense of the whispers - the Inquisitor does the same at that point, centering herself to understand the meaning of the elven words she's looking for.
From what I've seen of my IQ in relevant scenes after she drank from the well, Mythal's voice(s) seem to be similar to the song of lyrium, or the song of the taint (which, if certain speculations are right, may not be a coincidence); a subdued humming or a background white noise that is always there, at the back of your mind, but you get so used to that you don't even notice it anymore until it either gets too loud to ignore (like when Templars get overwhelmed by the thirst for lyrium), or you deliberately focus on (like when Wardens "feel" around for the presence of darkspawn). If that makes sense?
Oh that does make sense, thanks! Suppose that gives cullen and her something in common...sounds and voices in their heads...grrrrreat...

now to speculate on what will make it so loud she can't ignore it...hmmm... minus flemythal coming to claim her...too easy....