Shostakovich String Quartet No 8 - it quotes from the Piano Trio No 2. Do you know his Cello Concerto No 1, or the Violin Concertos?
I somehow think of the Kodaly Solo Cello Sonata as being in a similar vein, but I can't really say how.
There's some klezmer-type music in early Mahler - 1st and 2nd symphonies but most of all in Das Klagende Lied.
Following on from Schubert's Death and the Maiden, try Mendelssohn's F minor string quartet, Op. 80, written after his sister's death. Also Op. 44 No 2. Beethoven's Opp. 59, 74, 95 quartets are a better place to start than Op. 18 or the late quartets, I think.
For more of the roller-coaster side of Shostakovich, try Vagn Holmboe's Symphony No 8 (then 5, 3, 1, 10-13). Perhaps Gavriil Popov's first two symphonies, also.
Malcolm Arnold, Symphonies Nos 5, 6, 9. Robert Simpson, Symphony No 9 - like no other symphony I've ever heard.
If Shostakovich and Bartok quartets appeal, do try Elizabeth (Betty) Maconchy's 13 quartets, now cheaply available in a box set on Regis. I don't know them as well as I should.
And for something completely different: any music, but particularly motets, by Guillaume de Machaut (1300-1377).