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.
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.