It could have been the rain or just the plant growth/soil that caused them to push up. Throw enough soil to cover them by an inch or two. Then go ahead and throw the straw on top. How much are you talking? You don't want to bury them but a little covering wont hurt and should break down over winter.
To split them, just use a shovel down the middle. Now particular way to do it.
If you are transplanting 2nd years, it wouldn't hurt to treat them like first years. No doubt they will take a bit of damage during the move. Likewise, don't expect 2nd year harvest out of them, either.