Don't know if this answer helps you at this point but I will put this here anyway in case it helps someone else. You can still get the achievement even after missing one artifact if you meet the following conditions:
- You haven't already completed the quest Ill met by Moonlight (Daedric prince Hircine quest).
- You have missed only ONE artifact up to this point, which in your case seems to be the Mehrune's Razor.
In the quest Ill met by Moonlight, it is possible to get both Daedric artifacts and have them both count towards your total. You will be asked by Hircine at one point in the quest to kill Sinding. When you enter the cave, speak to Sinding, and help him kill the other hunters first. Then exit the cave and you will find Hircine waiting for you and he will give you the (restored) Ring of Hircine. Then go back inside the cave, find Sinding and kill him. Hircine should appear again and give you the Savior's Hide.
This extra artifact makes up for the missed artifact above and counts towards the total of 15 artifacts you need towards your achievement and provided you get all the remaining artifacts you should get the achievement. I am sure this works since I did this on my ps3 after missing another artifact like you did. I got the trophy.
Hope this helps someone.