Your headphones aren't the problem, the fact that you are bypassing your amp's speaker is(which is an integral part in the shaping of your sound).

As you've noticed is isn't that big a problem when playing with a clean sound, but when using distortion a lot of annoying high end does not get filtered out by the speaker...

Ideally what you should get is an amp simulator such a the Behringer V-amp, Line6 pod or Korg Pandora's box. These devises are specifically designed to be played through, for example, headphones.