I've dyed black Limba successfully without any heroic measures. But, as has been pointed out, it's a pretty open-grained wood so you can sand it until the cows come home and it's never going to get a level surface. You have to use a
grain filler. It's not a putty or anything like that, it's sorta like a high viscosity coating that you apply and squeegee off, then sand to remove any imperfections.
Usually, you seal the wood with a wash coat or a sanding sealer to prevent the wood from absorbing the filler, apply it, sand it back, and see if you're happy. With some woods, like Mahogany, Swamp Ash or Korina, you might need a second application to get the surface as level as needed for a fine finish surface.
Once you're there, you can stain if desired, or apply color coats, then clear coats. Finish sand and buff out, and smile at the sky.
This is a black Korina VIP I did a few years back that I dyed a dark red before applying clear lacquer...