Check out the Tindersticks, especially the first 2 records. Sometimes labeled I and II, but both just called Tindersticks. Great band. On that second record, Carla Torgerson of the Walkabouts shares a vocal on one of their best songs, and the Walkabouts have a few great records of their own that almost no one has heard. Like their beautiful covers record called "Satisfied Mind" from the mid 90s, or "Ended Up a Stranger" from just a few years ago. And someone else already mentioned Nick Cave. Everyone has a favorite, but maybe listen to "The Boatman's Call". Brilliant CD. Very nice sound too (and so is just about anything by the Tindersticks and Walkabouts). Johnny Cash had that string of "American Recordings" in the 90s, really great. I'm very partial to the first one.
Should I keep going? Definitely get a copy of Willard Grant Conspiracy "Regard the End". Very sad record, but it's a great one. And they live in nearly complete obscurity, but also you need "Sunset & Void" by the Flaming Stars. Think there's some talk around here about it from me and probably my buddy DLeeWebb. And Mark Lanegan, and Greg Dulli, and it really does go on ad on, even though The National is kind of a unique mix, with the droney, ambient backgrounds in some of their songs. I love the Cherry Tree EP best, but all of their stuff is good. The Cure's "Disintegration" has a lot of the same feel for me, especially the part borrowed from Joy Division. The Smiths from that same period fit in a little too. And sure, Leonard Cohen and Tom Waits (you do need Bone Machine). Most of the stuff above does tend to have the baritone vocals since that is key to the sound, though the Cure and Smiths would definitely not fall in that group, and the Walkabouts trade off between Carla and Chris Eckman.
Enter the Sandman. Lately I've been spending a bunch of time with those early 90s records by Morphine. Man, some great stuff. "Cure for Pain" has been rocking my car speakers for days. Might want to give that a listen if you aren't familiar. Great sound, fantastic set of songs, some of the same darkness that hangs over the National, that voice of Sandman, different mix of instruments with the amazing sax playing of Dana Colley driving much of the music. Love it. Probably no one else would ever compare them to the National, but who cares