No, this is not true. A counterexample is provided in Hirschhorn's book, Example 2.1.6 on page 36. See also the text on page 71: "Unfortunately, Example 2.1.6 shows that not all S-local trivial cofibrations need be $J_n$-cofibrations, and so there may be $J_n$-injectives (i.e., maps having the RLP) that are not S-local fibrations." I've modified Hirschhorn's notation to match the notation in the question.
This failure is also studied in depth in Barwick's paper "On left and right model categories and left and right Bousfield localizations." The point about maps with fibrant codomain behaving well is related to semi-model categories. Not every semi-model category is a full model category. Hirschhorn's example (credited to Bousfield) shows that you can't drop this condition in general.
Unfortunately, I'm on my way off MathOverflow, so if you want to talk more, just email me.