Can't Seem To Wrap My Head Around River Placement

[waldronate]

Because the islands are small, the rivers are small (the longest river in Hawaii is 28 miles long, according to its Wikipedia page). Because these islands are tall, rainfall will vary wildly from one place to the another, both because of wind and because of altitude. Climate conditions (including prevailing wind direction and available moisture) may vary over the lifetime of the island, so there might well be old channels for rivers that aren't there anymore. Certainly, some lava flows will adjust river drainages on the younger islands. The highest parts of the volcano are unlikely to have lots of annual rainfall simply because they are above most of the clouds.
The second attachment shows the relationship between trade winds and rivers in Hawaii. The first attachment shows some potential river flows based on a rough model of altitude based on your map (it most emphatically doesn't provide any model of wind or rain or temperature effects: it treats the entire area as getting a uniform amount of moisture per point on the map). Once you get an idea of the prevailing wind directions and rainfall amounts, you can possibly use some of the information on the first attachment to determine where rivers would be (prune away the parts that would be dry).