Looking at your samples, the answer seems clear to me: that's not grainy, that is, actually, out of focus.
Here's a 1:1 crop of your wide-open image:
It seems pretty apparent that the wooden sign is sharp but the dog isn't, and the appearance of the blur looks completely in line with what one would expect from out-of-focus blur, not noise or grain.
Outside of this crop, in the wide-open version, the trees are blurrier as well. I think that's a combination of decreased depth of field (as expected), and also increased chromatic aberration — an undesirable effect which is often strongest wide open. This is not particularly nice-looking blur, and might be what you are perceiving as grain.
Overall, in very small thumbnail size, neither defect is particularly apparent, but if you view or print larger.. not so nice. Since there is plenty of light, stopping down for more depth of field seems like a fine approach. Even the f/10 exposure gives you a plenty-fast shutter speed of ¹⁄₄₀₀th; there's nothing in the scene that benefits from 10× shorter. And, since the apparent depth of field is relatively large at such wide angle, with your subject in midfield,and when viewed at small sizes, wouldn't get strong subject isolation from background blur even if the focus were correct, so it's really not giving you much.
Also (and especially if you do keep the aperture wide, to see what it would be like with the blur more in the background), be more careful that the autofocus grabs the subject you are really interested in. (Note that AF areas are often actually much larger than the indicator dots in the viewfinder, so it's easy for AF to do something other than you expect.)
