1. Yes, it's an old-fashioned idiom meaning 'drunk'. It isn't really used any more.
2. She says "Oh aye, I ken fine what you mean".
"Ken" is Scots for 'know'. It's a word which the Outlander writers sprinkle liberally throughout the script to give it some local colour. You'll hear Scots characters in this series given otherwise standard modern English lines, apart from this one word. Be prepared to hear lots of "Ye ken?" and "Ye dinae ken?" if you carry on watching!