I am increasingly aware that my code in any single file can often span hundreds of lines quite easily and although I know the implementation might be sound, it still feels messy
We use stylecop. It helps a bit because it enforces a structure on your code and an order for what should appear where. Hence you can then find your way around larger files a bit more intuitively.
Separate your code into responsibilities. For each responsibility, define a single type. That is, follow the Single Responsibility Principal. Doing so will result in smaller units of code, each of which performs a very specific function. Not only does this result in smaller files, but also in better design and maintainability.
If your files are big because they contain a single class/struct that is big, then this is usually (but not always) a hint that your class is dealing with multiple concerns and can be refactored into a number of smaller, more specialised classes.
I go on the theory that if you cant see an entire method on one screen (i.e. you have to scroll), you should break the method up into further methods - either in the same class or when the code will get used more than once into a helper class.
To improve code readability: you can use the region block: https://msdn.microsoft.com/en-us/library/9a1ybwek.aspx . As for improving the structure and design of your code - consult some specialist books.
If I understand you, your main problem is that your forms end up being too big, which leads to the classes for those forms containing too much code, which is quite normal if your forms aren't very simple. The way to try minimize this is by using User Controls since if you move the controls to other classes, you also move the code behind to other classes.
It can sometimes make it a little more difficult to communicate between the controls, but that's usually more than made up for by the fact that the code in each class will be much easier to understand.