That helps.

First, I would suggest that you not worry much yet about the 350-yard shots. Bring your target back to 100 and get yourself comfortable there. I can only imagine your setup, if you're feeling any "spring" in the boards you're using as a pseudo-bench, I'd try something different. I have one of those inexpensive plastic folding tables that I've found surprisingly stable in the absence of a true shooting bench. Sandbag the forearm and the heel of the stock as Sheridan's already mentioned.

It's one thing to have an accurate rifle, it's another to do your part. The key is that every shot be much like the one that preceded it. So - the same ammo, the same bag arrangement underneath, holding your crosshair over the same point of the target (I recommend NOT using "bullseye" targets, get a square or diamond target intended for scopes), positioning of your cheek on the stock (and ultimately making sure your eye's looking through the scope the same way each shot), your breathing (many shooters let out half-a-breath, squeeze, BOOM, breathe, recover). You might google up the "eight steady hold factors" if you'd like more depth.

I know that some will advocate shooting five-round groups, though I prefer three. If you know you "pulled" a shot, note where the crosshair was on the target when the trigger broke ("call the shot") and re-shoot it. Get yourself comfortable at 100 yards and take the time to settle into the fundamentals before you march out to 350 (it'll save you some walking checking the target, too). When you're shooting consistent groups at 100 yards and satisfied, move to 200 and do the same.

Shooting for accuracy is a combination of very methodical and sometimes time-consuming acts. My wife wonders why I'm gone half the day when I'm "just going shooting." And you'd be surprised how quickly they'll come into play in "snap shooting" situations if you've practiced them correctly.

Good luck. Wish I had a back porch to shoot from!