Hi, Azure - the blue search box on this page may answer some of your question. I found tune and lyrics for "FINNEGAN'S WAKE" here (click), and lyrics and two tunes for "HIGH BARBAREE" here (click). Lyrics and tune for "HENRY MARTIN" are here (click). I couldn't find "Down at the Inn" or "Fallen Fighter" - maybe if you post a few words from the songs, somebody can help you find it.
I'd like to hear more detail from DougR's book that contends Charles Dibden wrote "High Barbaree." My guess is that all the songs I linked to are "traditional."
-Joe Offer-