The rit dye will work great. But just so you know too a red paint will work like a stain if you wipe it on and off and will seal the arrow at the same time.
It all depends on the look you want for your arrows.
If you get a red oil base paint and thin it down with a thinner of some sort it should work too, but the rit dye and alcohol is the cheaper way to go. Then just seal with a poly.
I don't recomend soaking your shafts but rather wipe on the dye. Moisture can worp the shaft and then you will have to straighten. Just use one shaft to try it on first, then when you get what your looking for do the same process to all the other shafts.