If you think of the wood's grain as straws and note
that a lot more dye will go into the ends of the straws
(the end grain) than will go between the sides of the
straws (the side grain), life will get a little easier.

Given that, if you dilute the dye you can always add
another coat to make it darker, whereas starting out
concentrated makes it difficult or impossible to make
things lighter.

So, if you can seal the ends of the straws by filling
them with something relatively colorless - like say
a blonde, super blonde or platina dewaxed shellac,
you can reduce the amount of dye the end grain can
soak up.

This assumes that you seal with shellac and then
dilute your dye with water (If you were to dilute
it with alcohol, which is what the shellac is cut
with, you'd reduce or defeat the shellac's sealing
function).

The other thing to consider is applying the dye
with an airbrush rather than a piece of clothe,
paper towel or brush - the latter acting as a
source of too much dye -- which the end grain
will suck up.

Water will raise the grain so a light sanding
with a fine grit before shooting the dye
will help.