I listen to some doom / sludge / drone metal, dark ambient, shoegaze...songs of which are generally meant to give ominous vibes so I see no need to draw attention to any specific songs under those genres. I'll name a few "entry-level" bands that might interest you: Jesu, The Austrasian Goat, Sunn O))), Swans.
There's an album called Hospice by The Antlers that never fails to move me to tears and has a very dark story behind it. The lyrics in this album are very strong and evocative and are accompanied by eerie instrumentals that completely jibe. If you're looking for something much lighter than metal but unique among typical alternative rock (a label with such ambiguous meaning, I didn't want to say "indie rock" though. a more accurate description may be "fuzz-folk"), I really recommend Hospice.
Here's a sample: warning foul lingo ahead
You can listen to the full album here if you're interested and have some spare time. You must read the lyrics along with it to get the full effect. I have shed many a tear over this album, I defy you not to upon listening.