I started Vaermina's Daedric quest in Dawnstar yesterday, I followed the guy to the tower and went inside, then it tells me to follow him but he's not there, the quest marker isn't even near where I am. So I go to my other file and try the quest, I complete it and get the Skull of corruption. But on the Character that the quest glitched on I have 10 of the artifacts already, so can I just collect the last 4 and I'll get the achievement or will I have to collect all of them again on another file?

You need only 15 Daedric Artifacts for the achievement and there are 16 in the game.

Yes you need them all on one character, and there is at least one that doesn't count.

Check wiki