You Definitely need Engineering Mechanics Books
You will definitely need good Engineering Mechanics Books as references books for the Mechanical PE exam. If you plan to take the Mechanical Systems and Materials PM session these books will be even more important!
During my time teaching I reviewed about every statics, dynamics, and mechanics of materials book out there. The books below are the ones I would recommend. I list more than one in some categories, but you only need one from each group. As always… use books that you are most familiar with when possible!! If you already have books from these categories that you are familiar with then use them! If you have older versions of these books that is fine also. The content usually does not drastically change in newer editions of books in these categories.
Statics and Dynamics
You can get books that contain both statics and dynamics together. The content is identical to that contained in the separate books, so the option of picking the individual books or the combined book is up to you.
Mechanics of Materials
If you plan to take the HVAC/Refrigeration or Thermal and Fluid Systems depth section you can probably get by without a vibrations textbook. You may encounter a couple of questions on the AM session covering vibrations, but the information you need would likely be found in a dynamics book. If you plan to take the Mechanical Systems and Materials depth session you may want to have a vibration book.