So far the hay has been great, I have even used oat and wheat hay with no issues. Seeds may sprout in the fall, but that is more biomass tilled in later. All the weeds I deal with now, are the same I'd be dealing with anyway.
Straw is good, but the coastal is easier to work with, much finer grass. I have never "inherited" weeds from using hay, especially if its a high quality horse type hay. I usually will try to get the second cutting or later if I can (less chance of weeds), but it usually fetches a higher premium. I bought a 600# round last year for $30 that was 2 year old wheat straw, even better now on the bottom.