How should a Christian feel inside? What does the Bible say about how a Christian should feel inside?

Should it be:


  Peace I leave with you, My peace I give unto you: not as the world
  giveth, give I unto you. Let not your heart be troubled, neither let
  it be afraid. —John 14:27
  
  For God hath not given us the spirit of fear; but of power, and of
  love, and of a sound mind. —II Timothy 1: 7


or is it more like:


  Many are the afflictions of the righteous, But the LORD delivers him
  out of them all. -Psalm 34:19


For example, at Church I get the feeling it is more the later one, of an anxious awaiting for the Lord to deliver people out of persistent trouble or stress, more than peace and joy inside. 

Is that wrong?
Your second quote isn't a "but" from the first. The whole point of Biblical staements about peace and joy is not that once you become a Christian, you will no longer have any problems in life and so you will have no reason to be anxious. Rather, it is that a Christian should be at peace regardless of what difficulties or trials he faces in life, because our hope is not in this life, but in eternity.

That's not to say that Christians DO consistantly maintain inner peace. But that's because our faith is weak. Just like we SHOULD never sin, but we DO sin all the time.