I am not a health professional but I would recommend that you get a second opinion. I would also recommend seeking the assistance of a Hematologist, this is a physician that works specifically with blood.
These are my specific recommendations. You have a dangerous problem if you are bleeding profusely like you have mentioned.
Being that it happened in the shower, you are lucky you didn't pass out and hurt yourself. Heat opens the veins. It is good for circulation. The heat would exacerbate your symptoms. In other words, the heat allowed for the blood travel better and quicker.
When you visit the Doctor's office, I recommend you argue for a full blood workup. They can test how fast your blood clots by doing an INR. Your blood dripping slowly could be affected by a multitude of conditions.
I am on a blood thinner for because I developed blood clots due to my birth control. What I have learned is that Vitamin K affects your blood's ability to clot. If you don't have enough of it, your blood won't clot well. If you have too much then it will clot a lot. There are a variety of herbs that affect the blood such as Ginseng and Gingko. Ginger also affects the blood. These are natural blood thinners from what I understand. But anyway, I am not going to say more because a) I am not expert I have just done research on my own and

you really need to see someone who can perform scientific tests and has a background in understand illnesses in depth.
I wish you luck. Don't hesitate on seeing a doctor. Waiting can be dangerous. Above remember, you are the advocate for your health, ask questions and think critically about what you are being told. Don't let the Doctor have the last word if you don't agree or if your situation isn't improving.