Here is a list of some of the songs I sing unaccompanied. Not all of them are slow and "haunting", though, but Tam Glen, She Moved Through The Fair, Quiet Land of Erin, Gartan Mothers' Lullaby, and others do fit that description. I'll put an asterisk * before the ones that I think fit the bill for your question.

Alice Flynn

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Kitty of Coleraine
Off to Philadelphia
The Mountains of Mourne
* The Irish Girl
Jackets Green
* The Foggy Dew (love song version)
* She Moved Through the Fair Click Here
Curragh of Kildare
Down By The Sally Garden
The Leprauchan
Bonny Bunch of Thyme
Ballinderry
Spanish Lady
* The Quiet Land of Erin
* The Galway Shawl
Where the Blarney Roses Grow
* Red is the Rose
* Rose of Tralee
* Lagan Love
* Paddy's Lamentation Click Here
* Fields of Athenry
* Carriag Donn
* The Gartan Mother's Lullaby * Farewell But Whenever
* The Wind That Shakes The Barley
* Easy and Slow
* Danny Boy
* Silent O Moyle
* Love Is Teasin Click Here
* Tam Glen Click Here
The Winter It is Past Click Here
* Country Lassie
* The Foggy Dew (a wan cloud was drawn...)
Spancil Hill
* She lived Beside the Anner
The Verdant Braes of Skreen
* Skibbereen
* The Fanaid Grove
* Connor's Return
* Green Fields of Amerikay Click Here
* Mist Covered Mountains
* Erin Gra Mo Chroí Click Here