When do we use 没(mei3) and 不(bu4) before verbs in sentences?
没 and 不 are fundamentally different.

没 means "didn't" 

不 means "won't" or "don't"

Actually your two sentences are perfectly normal Chinese sentences:

我没喝 means I didn't drink (it) -or- I haven't drank

我不喝 means I don't want to drink -or- I won't drink

A Chinese English Dictionary

没


  ADVERB INFORMAL
  have not or did not
  
  他来没来？——还没来呢。 
  
  Tā lái méi lái？ —— hái méi lái ne.
  
  Has he come yet?--Not yet.


不


  1 (used before verbs, adjectives, and other adverbs; never before 有) not; won't; not want to
  
  我不去。 
  
  Wǒ bù qù.
  
  I'm not going. or I won't go.
  
  他昨天不来,可是今天来了。 
  
  Tā zuótiān bù lái, kěshì jīntiān lái le.
  
  He didn't want to come yesterday,but he came today.


So the deal is: do you want to say haven't (yet), didn't (yet) = 没 -or- won't, don't (want to) = 不. Pretty much any verb can follow 没 and 不 everything just depends on what you want to express.

有 might be the only exception I can think of at the moment but that's been covered here:
Why is 有 (yǒu) the only verb that requires 没 while other verbs can use 不?