You're making the assumption that Future Dresden traveled to the past to change it, rather than Future Dresden traveled to the past to ensure everything happened the way it was meant to. Think Harry Potter and the Prisioner of Azkaban, where Potter makes the Stag Patronus that saves himself and Sirius from the Dementors.
If the situation here is like the one in Harry Potter, then the time travel is aligned with the temporal inertia, rather than against it, so there would be no temporal echoes in the past. Instead, the temporal echoes would occur near the time in the future where Dresden travels to the past, with the echoes being a result of the timeline changing if Dresden fails to do what he already did.
Ugh, this is why I LOATHE time travel. I am admittedly horrible at reasoning this out. But let me try...
Ok, only one of two people could have changed LC in our example. Now Harry or Future Harry. We know Now Harry did not do it because Now Harry has no memory of doing so and we don’t watch him do so (though I guess he has “forgotten” things like ordering his own death and his blasting rod, so this may not be a safe assumption, but let’s table that for now). Therefore, it is Future Harry which takes the actions changing LC (like in the example you used, it is Future Potter). If Future Harry does not act, then LC blows up.
If you say, well, Future Harry always was going to fix LC, so he is not changing the past, he is making it happen the way it was always supposed to, then
there would be no such thing as changing the past because every action Future Person takes is always the action they would have taken and the past is never altered [/u]. Yet we know from Odin’s speech that such a thing is possible, though difficult.