Dying lycra will pppprobably not work very well, you can try it with boiling water and add a little vinegar and it might stick a little bit but I think the best you're gonna do is tint it. Although if you're trying to go from off white to a little bit greyish, boiling water is probably fine? Always test first though obviously.
I don't think it'll actually prevent your panties from showing any better though. Seamless nude underwear/body liner is probably the way you're gonna want to go as far as thats concerned.