Don't Be Afraid to Break Things
Everyone with industry experience has undoubtedly worked on a project where the codebase was precarious at best. The system is poorly factored, and changing one thing always manages to break another unrelated feature. Whenever a module is added, the coder's goal is to change as little as possible, and hold their breath during every release. This is the software equivalent of playing Jenga with I-beams in a skyscraper, and is bound for disaster.
The reason that making changes is so nerve wracking is because the system is sick. It needs a doctor, otherwise its condition will only worsen. You already know what is wrong with your system, but you are afraid of breaking the eggs to make your omelet. A skilled surgeon knows that cuts have to be made in order to operate, but the skilled surgeon also knows that the cuts are temporary and will heal. The end result of the operation is worth the initial pain, and the patient should heal to a better state than they were in before the surgery.
Don't be afraid of your code. Who cares if something gets temporarily broken while you move things around? A paralyzing fear of change is what got your project into this state to begin with. Investing the time to refactor will pay for itself several times over the life cycle of your project. An added benefit is that your team's experience dealing with the sick system makes you all experts in knowing how it should work. Apply this knowledge rather than resent it. Working on a system you hate is not how anybody should have to spend their time.
Redefine internal interfaces, restructure modules, refactor copy–pasted code, and simplify your design by reducing dependencies. You can significantly reduce code complexity by eliminating corner cases, which often result from improperly coupled features. Slowly transition the old structure into the new one, testing along the way. Trying to accomplish a large refactor in "one big shebang" will cause enough problems to make you consider abandoning the whole effort midway through.
Be the surgeon who isn't afraid to cut out the sick parts to make room for healing. The attitude is contagious and will inspire others to start working on those cleanup projects they've been putting off. Keep a "hygiene" list of tasks that the team feels are worthwhile for the general good of the project. Convince management that even though these tasks may not produce visible results, they will reduce expenses and expedite future releases. Never stop caring about the general "health" of the code.