Can I just translate 犯 as often? I had this text: 我的性格比较好静，让我发言或接受采访我就犯晕。 

First I thought 犯晕 is a word, but I can't find it as a word.
I see 犯 can mean 'recurrence of something bad'. I also found:

英雄犯大错误。 Great men (often) have great faults.

In this kind of context, can 犯 be 'often'??
http://www.cantonese.sheik.co.uk/dictionary/characters/2546/
  
  犯
  
  /faan6/ /faan6*2 / ( jyutping)
  
  / fan4 /  (pinyin)
  
  [1] [v] offend; violate; go against
  
  [2] [v] invade; attack; encroach on
  
  [3] [n] criminal; convict; culprit
  
  [4] [v] suffer recurrence of (illness, bad habit, etc)


In the context of :

「犯頭暈」，「犯作嘔」，「犯肚痛」「犯口癢」and 「犯儍」,「犯」means " suffer recurrence of "(see [4])


「犯頭暈」= "suffer recurrence of fainting" 
「犯作嘔」= "suffer recurrence of vomiting " 
「犯肚痛」= " suffer recurrence of stomachache"
「犯口癢」= " suffer recurrence of open my big mouth."
「犯儍」 = " suffer recurrence of stupidity."


The word "often " describe an event happens frequently, but not regularly; while "recurrence" means something happens repeatedly, regularly. Therefore, The answer to your question is: "No" - 「 犯」doesn't mean " often",  it means: " repeatedly suffering" 

*「犯暈」is short for 「犯頭暈」


  Can I just translate 犯 as often?


You could translate 犯 as "repeatedly suffer" 

"I repeatedly suffer fainting every time they ask me to make a speech or take an interview."


  英雄犯大错误。 


It simply means :" The hero made (committed) a big mistake."