My personal impression is that at least on the level of foundational theory, Higher Topos Theory of Lurie is a good source. I guess this also explain the hard time you feel finding references: Gerbes seat very naturally in the context of sheaves of spaces (in this language this is just a sheaf of 1-types!), and I guess that this language has not fully penetrated into standard algebraic geometry texts yet, or any subject which is not modern algebraic topology, actually. However, the situation do get better with time, and I think that gerbes will appear more in texts soon (in particular, they are not out of fasion, just sort of get revised by $\infty$-category theory). For example, I personally almost finished a paper with a hole section for gerbes-based obstruction theory in etale homotopy, so I know there's at least one text on the subject that will be on the archive soon :-)

My personal impression is that at least on the level of foundational theory, Higher Topos Theory of Lurie is a good source. I guess this also explain the hard time you feel finding references: Gerbes seat very naturally in the context of sheaves of spaces (in this language this is just a connected sheaf of 1-types!), and I guess that this language has not fully penetrated into standard algebraic geometry texts yet, or any subject which is not modern algebraic topology, actually. However, the situation do get better with time, and I think that gerbes will appear more in texts soon (in particular, they are not out of fashion, just sort of get revised by $\infty$-category theory). For example, I personally almost finished a paper with a whole section for gerbes-based obstruction theory in etale homotopy, so I know there's at least one text on the subject that will be on the archive soon :-)