You can say either 'correct a mistake' or 'fix a mistake'.
As John has explained, 'fix' means to correct something which isn't working. If a student makes mistakes in their writing by using the wrong grammar or vocabulary, you could say that they need to 'fix the mistakes' by replacing the wrong words with the right ones.
On the subject of choosing words.... I admire your ambition to expand your English vocabulary, but 'oscillate' isn't quite the right word here. Things oscillate ( pendulums, temperatures and so on), not people. I'd say something simpler, along the lines of "I can't decide between x and y".