There's no other nutrients in the water...can it literally just grow off of light and no other food source?
There are always some nutrients. But when they're limited and light is excessive, algae copes far better than plants.
Starving the algae out is not an option. The plants starve too, so they sacrifice older growth and partially strip it of nutrients to support new growth. The old growth then decomposes. Decomposing organic material just happens to be algae's *favorite* food, attracting algae to your plants like a magnet. Parasitic algae makes the plant even more unhealthy, giving the algae even more food; in what quickly becomes a self-sustaining and hard to break cycle.
You'll have to reduce the light to eliminate algae's initial primary advantage. Ideally, you should add some ferts and CO2 with the goal of making the plants/mosses as healthy as possible, rather than a preferred food source for algae.
But like I said, once this cycle starts, it's harder to stop than to prevent from starting in the first place. If light/CO2/ferts are kept perfect and you're lucky, the hair algae will slow, stop, or maybe even shrink back, while the moss grows rapidly; allowing you to eventually discard the old infested growth. For most plants it's advisable to first trim away unhealthy growth and algae, but good luck doing that with java moss and hair algae.

So instead, I deal the algae an initial blow by another means:
1) Add 1ml Algaefix to 9ml water. Shake well.
2) Discard 9ml of the solution.
3) Add the remaining 1ml to 1G dechlorinated water in a Rubbermaid container. Stir.
4) Add java moss.
5) Set aside in a dark closet for three days.
6) Discard water. Rinse container and moss thoroughly. Soak moss in a copious amount of dechlorinated fresh water for a few hours to remove any lingering traces of Algaefix (it's very bad for shrimp).
7) Put moss back in an aquarium with good parameters for its growth that will lead to rapid recovery.
If all goes well, both moss and algae will appear to be completely unaffected. But the moss will be alive and the algae dead. The algae will decompose slowly over a week or more.