How to keep a big SOLID project manageable?

Having a big code mass demands some kind of principles to make it manageable. SOLID is one of the more used principle set. Even if it solves problems it appears to create another.

With a vary large codebase, following the SOLID model will drastically increase amount of classes, interfaces and injections.

The first problem is that when I find a object it is not always clear what I can do with this object or event how it is used.

For example if I want to display it, usually I would just type the object and hit dot and a list would show me what I can do with this object. With SOLID there would be another class "somewhere" that handles display styles. How do I find this class? How do I know that it even exists? References sounds like the way to go but it could show over 100 ref and a lot of them may not have anything to do with functionallity of the class.

The second problem(might not be a SOLID problem) is that when using injections it can becomes unclear what code that really is executing during a specific flow. Injections uses interfaces and there can be several classes that implement this interface. So when investigating a specific flow and we encounter a injection we need to dig in to what object that really is injected to know which code that is executed. In a large code this can be vary time consuming to find.

How is theses problems solved?