How serious is blood after a bowel movement?

I am happy to hear that you scheduled an appointment with your doctor. Blood after a bowel movement is never "normal"- it is a sign that something is going on. See also:
Where does the blood go that has to be replace due to chemo treatments? Any bleeding from the rectal area should be checked out by a health care professional.

While the small amount of blood on the toilet paper after hard BMs probably is from hemorrhoids, the clots you describe are more concerning. Your doctor will probably refer you to a Gastroenterologist to see what is going on with your chronic constipation and bleeding. You will probably need a colonoscopy to see what is happening inside your rectum and colon, so they can figure out where you are bleeding.

I know that you are concerned, but you are doing the right thing by getting it checked out. I am sure the thought of cancer has crossed your mind, as it is a possibility. But, your bleeding may be from hemorrhoids, polyps, or diverticulosis as well. Hope all goes well, and good luck.