From the excerpt that you have posted, it seems that the passage is written in a creative style that may not follow grammatical rules exactly to the letter. The first sentence is not strictly grammatical according to formal rules (though it is quite acceptable in the style in which it is intended). Yes, "higher" means that they climbed higher (or just were higher); "dry" probably means that the ground is dry, in contrast to the "rills and freshets" of earlier.
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