My mother has factorvblood coagulation disorder. I haven't gotten tested yet. My period is dark brown and clumpy/stringy. Can this be a sign I have it?

Hard to say. Factor v leiden as you mentioned is a clotting disorder making you more prone to them. This could be made worse with contraceptives. Seeing as your mother has it, i would advise testing. It is usually discovered after clots are formed in the deep vessels of the leg, lung, or perhaps even recurrent miscarriages that prompts investigation.