Well For moss to develop, it needs three conditions – shade, temperatures between 32 degrees Fahrenheit and 70 degrees Fahrenheit and moisture. 

Do you have anyway of heating your basement? Like renting an industrial heater? Cause then you could kill it off that way. But as I said earlier a hard wate store might have sprays and chemicals (harmless to you but deadly to moss and mold(moss by itself is actually harmless but when there is mold the spores from it can make you sick)) that can help

First though, I would simply use bleach. Wateured down, you can spray it on the walls, floors in between crevices and such, as well as pouring it down drains and water outlets. Bleach is wonderful for Moss and mold. This is the cheapest option, so id go with that first-make sure though that you have a mask on, right now it sounds like you have mild forms of mold sickness although I don't like the fact you are having bad headaches, so wear a mask when you go to spray the basement, bathroom, drains, basically anywhere where it is damp and dark, and do it weekly until you move. If there's jo improvement using bleach,
Then move onto the other options I gave. But try the bleach first. Get a spray bottle, use 1 part bleach for every 3 parts water (I think that's right but it's meant to be 1/3 bleach in anycase with water), put the mask on and get spraying! 

Also, when you get this headaches, go outside, or open a window and stick your head out. Take some deep breaths, and have a glass of water