With stomach problems you can't take aspirin either, that's my choice if the risk factors of clots are not high.
As I understand it, AFib increases the risk of clot formation, but it does not mean one will get a clot when an AFib episode occurs. The prognosis is simply the risk is higher, but still low. I've seen a number that permanent AFib (such as I have) increases my chance of a clot/stroke to about 3% a year, cumulative I believe. So, over a period of years my risk is high, but again I am in AFib all the time. I take warfarin for that reason and I tolerate it well, I have no stomach problems so far. I also take 1/2 a standard aspirin a day.
Some folks do well on Coumadin, others can not tolerate it. If you are afraid of Coumadin, but want something that will help thin the blood, I would suggest the herb "Ginger" this can be purchased in pill form, and it has the same blood thinning properties as aspirin WITHOUT the side effects. Since it appears your A-Fib is not constant I think I would go this route first, oh yes, also, a good, liquid fish oil would keep the blood slippery and clot free. I take one by Carlson, I am used to the taste now, and believe me, it helps.