Water

Lost dropped to her knees at the edge of the creek, cupping her hands to test a sip of the water. Experience with shorelines left her confident that the salty taste was just the sand and not the source. Splashing a bit on the back of her neck, she stumbled up towards the treeline to investigate where this was coming from. She tested another sip under the first leaves of the looming plants - fresher, yes, almost fit for drinking. Before venturing further, she unsheathed the sword that had been hanging on her hip all morning. She had yet to meet any wildlife beyond the local insects, but she was of sound enough mind to recognize the possibility of others who laid claim to this water.
If the brush nearby was thin, a spot close to the creek would be a grand place to settle in. Sword at the ready, she continued to trace the creek inland a bit and survey the area around it for a suitable shelter location. She saw a promising patch not far in from where the trees met the shore: two of the palms had fallen of late, but away from each other. The space between them... Growth was thin, as the palms had kept other plants at bay by their mass until what seemed to be recently. It was fairly flat, not too vast, and the creek could be heard over the wind rustling the leaves about her.
Yes.
Here.

Is that...? Could it...? Maybe...?

Lost continued to follow the shoreline southwards. The sun was beginning to steal the moisture from her body as she pressed onward, seeking unsalted water. The ocean, as tempting as it may seem in the heat, would be a fatal choice. Forward...
The sunlight and exertion was addling her further than her already hungry belly and sleep deprived mind already had thus far. She knew better than to expose herself, though - the direct light was worse than merely its heat. Onward...
As she came out from around a blind bend of the shoreline, something looked awry. Something about the sand did not match the other sand... It looked darker... It looked more... reflective? She couldn't put the pieces together, so she simply continued towards it, hoping answers would present themselves. She didn't care to ponder; she was too weary, and whatever oddities of the sand would become clear once it was underfoot.
There was a subtle shift in the sounds of water beside her. The waves sounded the same, but there was something else. What... What could it be...? The heat and thirst were making Lost dizzy, and this new sound confused her. Step... Another step...
The dark, shiny sand was just up ahead, now. Curious, how that stripe ran from the ocean to the unending wall of trees... Realization struck her rather bluntly as the small creek came into focus before her.

Southward

Deep breath... Lost headed southward along the beach, still wary of what lurked beneath the trees. Hopefully any springs about her did not rebury themselves before reaching the shore; if that were the case, hope of survival is all but lost before having a chance to glow within her. The chills of the night before left her with little energy to press onward, but it was not as though she had much to return to - the lean-to could be rebuilt elsewhere; anywhere... There seemed no shortage of trees about.
The scenery changed little as she moved forward; this could be a good or an ill sign - it was to early to tell for sure. Eyes alert but hazy looked forward down the shoreline, seeking signs of water besides the abundant ocean. The shore curved a bit this way and that, sometimes helping and sometimes obscuring her sight of what was ahead. The gnawing on her stomach was a distraction but not of grave concern. No, that which was all too abundant but of the wrong sort was what she needed: water.
The only thing reassuring was the consistent sound of the waves around her. This island was unknown, but its kind was familiar. The waves that she had rocked in for much of her life were still there beside her; not everything familiar was taken from her by the pirates; they could not steal away with her memories of the sea.

Morning

With the weariness felt by one left isolated on a presumed deserted island, Lost rose to face the dawn. The haze of sleeplessness blurring her thoughts, she tried to focus down on the most pressing needs. The chill to her core said shelter; the emptiness of her gut saying food; her rational mind saying water and tools.  However, water would require exploration for a spring; perhaps that should wait until she had weapons beyond her short, rapidly blunting sword... Perhaps not.
She walked out to the shoreline, allowing the surf to lick her feet in return for a wider view - were there any rivers or rivulets wending their way to the ocean? None in sight...
A new choice to be made - north or south? From the rise of the sun, her berrings were clear - she'd been left on the eastern shore of...where ever this was. Which way to fresh water? Would there be any water even if she circled the whole of the island? Would one direction hold water and the other beasts? ...Deep breath... No use fretting over the unknown; this whole island was an unknown. Progress would be impossible if she only stayed within the confines of what was familiar to her.
... ... ... South it is.