Is this statement:
the letters have been troubling me since over a year
grammatically correct? Or should I instead use:
...for over a year
"Since" is a word that is often mistranslated from other languages. Be very cautious in its application.
In this case "for" is correct. "Over a year" is the sticking issue here. It's not a time but a period. "for" implies duration, i.e. covering a period of time. On the other hand, if you say "since last year" then you're picking a specific time and that would work.
In this case you can use only for over a year. You could use since if it was the last year.