J.A.
Hi K.-
I have a clotting disorder, too, and have to have non-estrogen birth control. I got an IUD after my girls were born. Until recently, when I restarted warfarin therapy, I didn't have a period. Now it's started back up, but it's very light and short.
I have the mirena IUD, good for 5 years. I like that I don't have to worry about remembering to get a shot or take a pill. I also like that, even with the blood thinners, I don't have a super heavy period like I used to get.
Good luck!