Look at what you have achieved, not at the obstacles yet to be overcome.
Rather than worry about too much luggage, or how you will be able to perform a certain task, check back over this thread and see just how far you have come already. Take pride in your achievements.
As Dave said, just get on the road and stop worrying about the detail. It will all fall in place. When faced with the necessity to do a certain task, the mother of invention always comes to the rescue. You will meet other folk at hostels and campsites, and possibly get to chat with them, and you will all share what your challenges and plans are. You learn from others, and before long you will be teaching other newcomers to the task, and showing them how you overcame a certain challenge. Washing dishes is not the only one you will face. [I have always dried my washed dishes... never wasted more water on rinsing them. Use good quality dishwashing liquid, sparingly.]
What you really need to do now is hit the road. If you think you still have too much luggage, then weed it out as you go. Things you find you are not using at all, clothes you find you are not wearing, donate them all. Anything you wish you had, but did not think to bring, keep looking in charity and second hand shops till you find it.
Relax and enjoy the journey through your new life.
Lifey