Your fears are regrettably well-founded. Unlike mathematical logic, the philosophy of language is not so clear cut and remains to this day a deeply controversial topic. There are a dozen schools of thought on the philosophy of language, both classical and contemporary. It is a fundamental and unquestionably significant field of philosophical inquiry, but unfortunately what appears as a stream of simple concepts that should be amenable to science soon reveals itself as an inconsistent web of increasingly intricate questions and facts rapidly give way to opinions. The philosophy of language divided, and continues to divide, the analytic philosophers—who are known for their predilection for rigour, valid method and proof.
While I would love to hand you a shiny Cambridge or Springer text in the philosophy of language, I am afraid there is no definitive work. There are commentaries and historical discussions but these will not suffice at all. What I recommend instead is this: if you are sufficiently intrigued and if you have the time to spare, read through the texts I have listed below, which come straight from the field's most significant philosophers themselves. For if it is not possible to find an objective and rigorous account of something, one must build an internal consensus for themselves, generating an objective account from a rigorous treatment of the intersubjective account.
Some essential texts
Gottlob Frege - On Sense and Reference
Frege, the first to undertake the project of founding mathematics in formal logic, also made this important contribution to the philosophy of language: that meaning can be divided into sense and reference, and though he was not the first to see it, his discussion is comprehensive and analytical.
Bertrand Russell - On Denoting
Seminal work in epistemology and philosophy of language in which Russell rejects Frege's distinction of sense and reference, arguing that propositions in language must have a referent and that sense and reference do not exist independently. From here, Russell builds up to an alternative notion: two different kinds of knowledge, namely knowing x from a description of x and knowing x from direct acquaintance with x.
Ludwig Wittgenstein - Tractatus Logico Philosophicus
Wittgenstein's first published work, during a time when he, like Frege and Russell, aimed to root mathematics in set theoretic logic and reduce the world to a propositional calculus. This text not only discusses the limitations of the logical atomist project, but also develops an innovative picture theory of reality, and throughout the nature of language is utterly central.
Ludwig Wittgenstein - Philosophical Investigations
Wittgenstein's magnum opus is a selection of notes organised into a book that discusses the notions of grammar as forms of life, of language-games and of rule-following, all hugely influential ideas in the philosophy of language, showing that meaning arises through use, not through an elementary pointing-and-naming process.
Willard Van Orman Quine - Word and Object
Quine expounds his behaviourist picture of the world through a discussion of language acquisition, of the difficulties encountered in translating between languages, and of philosophy of mind.
Saul Kripke - Naming and Necessity
Principally a logician working with modal logic, Kripke was also interested in the philosophy of language and Wittgenstein's ideas regarding a grammar of life, and made an effort to resolve the rule-following paradox that Wittgenstein revealed in his Investigations. In this book, Kripke develops his ideas about how meaning is shaped by a causal frame of reference shared by a group of speakers.
Donald Davidson - Inquiries into Truth and Interpretation
Davidson's enormously influential philosophy of language in a single volume, pioneering truth-conditional semantics and discussing the nature of languages and grammars as finite in form, i.e. finite in rules, even if infinite in potential expressions.
J. L. Austin - How to do Things with Words
Further development of Wittgenstein's language-game concept, looking at how these games break down to different types of spoken events, discussing the significance of the revelation of these types.