Julian Schwinger was a great, careful physicist.
But I think it's an abuse of terminology to use the term "theory" for the collection of insights included in his "source theory". Instead, it was really a pedagogical approach to talk about quantum field theory. The pedagogical approach tried to avoid quantum fields (operator-valued functions of spacetime).
Instead, it focused on the transition amplitudes in the presence of sources – they were really the generating functionals. The coefficients in the expansion of this functional are the usual Green's functions. So Schwinger's approach to QFT was really an approach giving a prominent role to Green's functions.
None of these technical constructs has become outdated. In QFT courses, we still teach Green's functions and other things. They're just not being associated with Schwinger because he hasn't really discovered the Green's functions (or other concepts we routinely discuss). And virtually no one shares Schwinger's opinion that one should work hard in order to avoid quantum fields (operator-valued functions of spacetime).
I would even say that Schwinger was describing QFT in this way to become the father of "another picture for quantum theories" that would be analogous to Feynman's path integrals, for example. But I don't think that he really had a new picture. Schwinger's approach was a meme that was supposed to become popular but it never did. Much of it may be viewed to be exactly equivalent to the quantities computed by Feynman's path integral.
At some moment, Julian Schwinger's life-long work on source theory would be considered boring and de facto inconsequential by his Harvard colleagues so he left Harvard for UCLA. I think it's fair to say that it's still being considered inconsequential today. Note that his first paper on "source theory" was Schwinger's most cited paper but it was really because of the particular calculation he did there, not because of the formalism he started there and spent his life by promoting it.
It is not clear to me what one could possibly mean by the statement that a calculation in string theory used "source theory". Source theory is just another way to talk about the very same concepts and calculations in quantum field theory that we normally discuss. It's not an entirely new set of methods or insights about the physical system (quantum fields). And even if a string theorist was using Schwinger's terminology, it probably had no impact on the calculation at all. It would just make his wording less comprehensible to others. The paper could have been easily translated to the prevailing language that avoids Schwinger's focus on sources (and his largely irrational desire to avoid quantum fields themselves).
If your professor can teach you QFT using Schwinger's "source theory", good for you (I also remember a textbook that was using it, and it wasn't even written by Schwinger, but that book was very old, too). It may be good for the physics community if someone still knows about the other directions of thinking and interpretations that once existed. If your professor knows source theory well enough to produce students who "see through" using Schwinger's old lens and who can still understand the contemporary particle physics, it's great. But if you can't learn QFT in this way to do the other things that are needed and that others are apparently doing with ease, then just forget it and join those who consider "source theory" to be a nearly forgotten, idiosyncratic way to talk about the very same thing that may be dropped without a loss of generality.
Every insight about perturbative quantum field theories could have been translated to the package of Schwinger's source theory except that this "language" has become sufficiently unpopular so that most of the insights, even rather elementary ones, have never been framed in that way. I actually don't know how one would talk about simple things like "dimension-6 proton decay operators" in his approach because he wants to avoid operators. It's not really the preference for sources but rather his hostility towards the normal operator-valued fields that makes Schwinger's approach almost unusable in practice.