In agreement with yourself, I find it frustrating and annoying when an individual is called a Project Engineer or Program Manager. It seems to be buzz word for the masses, as everyone wishes to be a leader.....
Do a web search for PMI or Program Management Institute. This an organization which trains individuals for a certificed PMP (Program Manager Professional). Some cities will have a PMI group; which will help you learn new skills, and network with other individuals who are program managers. I have no suggestions for books with good engineering examples. Sometimes a Project Manager is not evualated on how well they did the project, but how happy the client is, and if the project made money.
In the corporations I have worked at (both industrial and automotive), the project engineer is the person who handles the project from time the contract is signed to the time the unit leaves the building. Sometimes, resources report directly to you, and other times you need to manage people who are not reporting to you. The individual is responsible for the client correspondence, design, procurement, and debugging of unit. It can all be overwhelming if you do not have good organizational and time management skills.
To start small, develop a system which helps you track the project. Once you prove to other project managers that your system is working, saves you time and headaches, then they will quickly pick up your system, or your boss will impose it on them.
From your comments it sounds like your company does FMEA analysis, but no one is assigned to follow up, and once the widget ships, there is nothing driving the solution forward. This may be an ideal situation to sort through all the outstanding FMEA, and create a new project!

Resolving all the outstanding FMEA, before one of your clients calls to complain.
Babyyoda
Fellow Project Engineer