After playing BANG! for a few years, my group realized similar issues to yours.  

A few we've implemented:

 - **Fog of War**:  For the first few rounds, players get a bonus to their defense that decreases by 1 each turn.  Fog of War is equal to [# of players - 4], up to maximum of 3. 
  - Improves Sheriff survivability in early rounds.
 - **Grave Robbing**: Once all Outlaws have been eliminated, all players that are not the Sheriff draw a card.
  - Improves Renegade chances, especially when playing with more than 1 Renegade (which I recommend past 6 players)
 - **Man of the Law**: When the Sheriff dies, the Deputies reveal themselves.  The Deputy next in turn loses 1 HP and becomes the new Sheriff.  This life loss cannot kill the new Sheriff.
  - Improves chances of The Law dramatically.

For 6 players, the Fog of War variant should work fine to mitigate the burst-damage of 3 outlaws.  If you feel otherwise, replace it with the Man of the Law and the Grave Robbing variants. 

When playing 7 players, I'd recommend using a second Renegade instead of a Deputy, along with the Grave Robbing variant, as this allows Renegades to invest heavily into dealing with Outlaws while still having enough resources to deal with the 2nd phase of the game.