Hello,
Each creek, stream, river and water tributary is within a watershed. And you pointed out you were hiking at a certain elevation. Water flows downhill due to gravity. And the visible water (visible streams, rivers, creeks) are not the only sources of water that flow downhill. There are underground channels of water (groundwater) that can add to the downstream water. And if you were to map a watershed, the different channels flow down typically to a collection area (river, lake or large stream) before they flow to their main channel (large river, lake or ocean).
In regard to the question why don't the rivers dry up, it is because there are other sources feeding into them that you aren't observing or aware of. Again, watersheds cover a very wide area and typically follow gravity to flow downhill to a collecting channel.
Hope this helps answer your question. Best, Lisa H.