#4 is right: Birdsong is a powerful book, well worth seeking out.

I enjoyed it, but Faulks wasn't there and I think unless you were you can't really appreciate what it was like, and therefore portray it accurately.
Maybe you should look for WWI non-fiction, if that's your criteria. Lots of people were there and can't write about it convincingly. Fiction's meant to be convincing, not accurate.

Pat Barker's (admittedly not there) Regeneration Trilogy is superb and certainly gives the convincing impression of being there
Fictionwise, not sure other than the ones you already mention.
Plays - Journey's End
Poets - Sassoon, Owen
There is a book about Rivers - the famous psychiatrist and his patients - can't remember anymore about it but a good bookshop should be able to look it up.

Regeneration Trilogy is brilliant. So is Another World by Pat Barker.
Under Fire: Henri Barbusse - follows the French 6th battalion, based on his own experiences.
In Desolate Heaven by Robert Edric - not written by a witness, but still really good.
The Roses of No Man's Land by Lyn MacDonald - about nurses, but still very graphic with lots of letters and extracts written by the people who were there.
A Month in the Country by J.L Carr - about a guy coming 'home' from the first world war.
Mud, Blood and Poppycock by Gordon Corrigan - a 'history', with loads of extracts from diaries and things.