Before you assume that the books are illegal, you need to inspect the terms of the agreement between author and publisher, if any. Very many publications are completely legal and give no royalties to the author (95% of mine are royalty-free). An author may transfer copyright to a publisher, or he may license its publication. An actual transfer needs to be in writing, but a license can be implicit (and often was, again most of my earlier papers). When you see a copyright sign and the name of the publisher, that does not mean that the publisher has the copyright to the entire content: it may only have copyright in the arrangement of works, and individual authors control their own works.
A truly illegal work would be one where the publisher takes the author's work and copies it without permission. Publishers don't break into authors' property and steal manuscripts. The burden of proof does rest on the publisher to show that they have permission, which could be in the form of letters saying that a work was to be published in some compilation; corrected page proofs; and various other ways of signalling agreement.
A likely source of truly illegal copying would be re-publishing previously published works, e.g. Smith licenses the right to re-print a work from original publisher Jones. In that case, (1) it matters whether Jones got a transfer of copyright, vs. relied on licensing and (2) whether in the latter case the original license prohibits sub-licensing. If Jones relies on implicit licensing, they would have a hard time establishing that sub-licensing was part of the implicit sub-license.
Supposing that your copy is truly illegal, there is no way to make it legal, because the illegality is copying without permission, and the copying is a fait accompli. Permission would have to be granted by the rights-holder to the publisher: the owner of the book does not need permission from the author to own the book, they need the permission of the publisher or publisher's agent (e.g. the bookstore). And you have that, in the form of a standard sales contract (assuming you didn't steal the book).
The foregoing is general stuff about copyright: there is a second concern not about the book, but the right to publicly perform. 17 USC 106(4) gives the author the exclusive right
in the case of literary, musical, dramatic, and choreographic works,
pantomimes, and motion pictures and other audiovisual works, to
perform the copyrighted work publicly
Supposing that you don't just want to own the book, you want to use it in performances, then you need a license to do so. You might directly contact an author and attempt to negotiate permission to perform, but this would be quite a burden (especially if the author is deceased), and also not foolproof. The problem is that the author may not actually know what the legal status of his copyright is -- he may have just signed a form without thinking about what he was signing, and might have transferred his copyright to the publisher.
Regardless of the license / transfer question about the author and original publisher, the fact that you have acquired a copy of the book does not mean that you have a right to publicly perform. Presumably you know that, and have a performance license.
ASCAP claims that it rewards songwriters; BMI and SESAC do likewise. I have no idea how true that is and especially how vigorously they trace the legal history of a work. You can identify individual authors and directly compensate them for their work (assuming you can locate the author; then figure out a just reward). Ultimately, there is no bullet-proof way to know that you have permission to perform, but it you're concern is with rewarding the producer, you can reward the producer.