[Updated] CUP have just emailed me information about this new book, coming out in March, An Invitation to Model Theory, by Jonathan Kirby.
The contents list and the description of the aim of the book look very promising: “[T]raditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics …”.
Daniel Nagase notes that there is currently a late draft linked at Kirby’s webpage. That link might not survive the actual publication of the book, so if you want to download to take a look at the book, hurry, hurry, while stocks last!