My 3 year old crown split, and the warranty has expired. Insurance won't cover. Should I look for a new dentist?

While it is unfortunate that the crown broke, it is not likely that fault lies with the dentist. There are so many factors involved that "blame" is hard to apply.   Finding a new dentist is up to you, but if trust is broken then perhaps you should move on. 

It is up to individual dentists how long they warranty their work.  The difficulty of each case varies and the amount of stress applied to restorations is significant.  Having said that many of us warranty our work beyond one or three years for patients who are regular with their follow up visits.