No, there's nothing wrong with your amp. Only, remember that the tone of any guitar amp is shaped by all of its three main components:
- The preamp section, including the gain, tone controls and channel selectors (if any). This is usually the only section where you can tweak the sound, but that doesn't necessarily mean it's the most important for the final result! Most of the overdrive is generally produced in the preamp stage, but the tone shaping actually is more a very rough pre-processing step. It does not attempt to create a signal that sounds good by itself, but a signal that will make the complete amp sound good.
- The power stage, which takes the preamp's signal and boosts it enough so you can be heard in a band. Modern solid-state power amps are generally designed to not influence the sound much (except for its level), but the good old all-tube amps actually take a lot of their character from the secondary overdrive in the power stage, as well as tone-shaping through the output transformer and nonlinear properties of the power supply (rectifier tubes). Therefore, these amps only sound “proper” when actually played loud.
- The cabinet, i.e. the actual loudspeaker which translates the electrical voltage signal into pressure waves that you can hear. Again, modern PA- or studio speakers normally aim to colour the sound as little as possible, but that's by no means true for guitar amps. In fact, the cabinet's response is quite the most complex part of an amp's sound character – it may not add as much dynamic compression / nonlinear distortions as the preamp, but it has a highly interesting frequency response, with a strong cutoff at high frequencies, substantial resonances and general smearing of the transient response. All of that tends to make the sound a lot smoother, and is particularly helpful to prevent distortion from becoming a single nasty piercing rasp.
Now, if you're using the amp with headphones, what you hear is normally straight from the preamp section. So it does come with overdrive, but without the main important tone shaping that would actually make the overdrive sound good!
If you want to practise playing through distortion quietly, use some digital amp simulator. Those imitate not only the preamp overdrive but the full signal chain, including cabinet simulation. Of course, it's debatable whether digital simulations are good enough to replace real amps live or in the studio (IMO the better ones are, nowadays!), but for practising they're definitely ok.