Macroeconomics Textbook on New-Keynesian models
Michael Woodford's book Interest and Prices, while it may not be explicitly New Keynesian, may have some of the rigor you're looking for applied to this class of models.

A more direct alternative would be New Keynesian Economics edited by Mankiw and Romer. While it's a collection of papers not a textbook, if you're looking for underpinnings of New Keynesian models this'd be a good place to start. Also "The Science of Monetary Policy: A New Keynesian Perspective" by Clarida, Gali and Gertler.