I use Warfarin, real inexpensive, I think it is the same as coumadin. I have never had dizziness with from Warfarin, beta blockers, yep they can cause dizziness.
Careful control of the medication range should prevent any bleeding problems. There too, lucky me, I have never had a bleeding problem... yep, when cut I bleed more freely, I think, but have never had a problem stopping with pressure and a tight (pressure) bandage.
My cardiologist told me it is expensive and has no antidote. With Coumadin, in the even tof an emergency, you can be given Vitamin K or plasma. I have been on Coumadin 2-1/2 years and check my INR at home. I have not had any problems with it. I do have to keep my vitamin K intake consistent and many medicines interact with coumadin. Good luck.