Well I'm sticking with dorian...

Frankham says that in E the C# doesn't appear in the melody. It does.

On the third 'drunken sailor' the notes are B C# D B

Have a look at the relevant section of the essay I linked to above. It should make things clearer (although the article admits that it's not 'strongly' dorian).

In terms of modes in general, the starting note has nothing to do with it. It's the home note (tonic) that the tune resolves to that is the important one. Virtually all folk music ends on the tonic, so the last note of a tune is the one you should be looking at.

The MODES FOR MUDCATTERS thread linked to at the top is also worth a read.

Hope that helps, and feel free to ask any more questions

Ed