As Robert has already pointed out, this is a modern cover of Roud 1286. The inimitable Traditional Ballad Index has good notes on the song; of particular interest is one note which reads:
Captain Grant, an admitted robber, escapes from Edinburgh jail where he was sentenced to be hanged "For sheltering M'Kay (emphasis mine) Although I had no hand in that robbery
Based on this, I'd ament the line to read "I was sentenced to be hanged for sheltering M'Lay" --likely a misheard version of this M'Kay line.
The line on "hungry and dry" is definitely a reference to drinking -- one version of the text (link) follows that line with "You take them to some public house [ie, pub] and there their wants supply," while a second (link) sees Captain Grant take his unfortunate friends to an alehouse.
I haven't been able to find out who "Lord Natty" is meant to be. While the story of Captain Grant is based on the execution of Jeremiah Grant in 1816, he was arrested specifically for the robbery of Thom Cambie, Esq. -- I haven't found any lords involved, and after skimming through a book of notable English criminals of the period, I never saw a "Lord" whose name fit the bill. Other versions I linked previously have the line before Grant's final return to jail as "To the valiant bold hero called the Man taker up" or "To a valiant young hero a man ..taken.. up" (link). Possibly it's another corruption of oral tradition, like the M'Kay / M'Lay line.