Tiger - "Not sad, but teary anyway" is an interesting category. I find I'm much more likely to cry at an uplifting song like The Mary Ellen Carter than an ostensibly sad one. Add the effects of over-exposure (and I find some songs get over-exposed quite quickly) and an awful lot of the songs on your list leave me dry-eyed.

Lal Waterson's Child among the Weeds has me in bits if I ever try to sing it, even around the house. Sydney Carter's Every Star Shall Sing a Carol likewise, not to mention Friday Morning with its devastating chorus/punchline -

"It's God they ought to crucify instead of you and me,"
I said to the carpenter, hanging on the tree

Peter Bellamy's Us Poor Fellows and The Leaves in the Woodland both need a few run-throughs before I can complete them.

Traditional songs don't affect me like that when I come to sing them. There are some powerfully affecting lines in some of them -

Send for the doctor although it's too late

Had I lived I might have been clever

And every stitch she put in it, the tears came trinkling down

I have lost a sheath and knife that I'll never see again

But they don't hang about - you sing the line, you tell the story, and then you move on to the next line and the next bit of the story. When I sing The Unfortunate Lass or The Trees They Do Grow High I'm in the song, with no chance to reflect on it until it's finished; I don't find that to be the case with contemporary songs.