Dying faux fur can be done, but it's not the easiest job. The difficulty is that it's a synthetic material & so doesn't absorb the dye too well.
I prefer to use paint. Cheap acrylic stuff is fine, there's no need to waste your expensive modelling paints!
The best way I've found to apply it is by hand. I mix up the paint, dip my fingers into it & then rub it into the fibres trying not to overload the fur. You want it slightly damp, but definitely not wet. That will cause the fibres to stick together & ruin the effect.
Another way is to
airbrush the fur. Whilst this doesn't get the paint into the full depth of the fur, it's useful for adding effects like dry grass using a tan colour.
HTH