ok, this is a common problem area that seems to catch quite a few people out, (caught me out around 20 years ago when i was first tinkering with car audio).....
the distortion your talking about is down to the headunit....
as stated previously a speaker merely reproduces sound in accordance with the signal it recieves, if this signal is 'clean', then the sound will be 'clean'....
car manufacturers have recently discovered that nice sounding audio systems are a good selling point, so more and more are fitting half decent speakers and some are even amped seperately, (Bose systems for example),....
lets look at your Sony unit;
its probably fused at 10amps, this 10amps is basically shared between;
cd laser
disc motor
cd laser carriage
unit illumination
flashy graphic display panel
internal amplifier
plus any aux units like cd changers or I-pods
as you can imagine, the power available to the internal amp is fairly limited, the higher you turn it up, the more likely voltage drop will occur within the headunit and esp the internal amp....(you will see this as a dimming display panel)....
some headunits will go full volume with no clipping through either the pre-outs or internal amp, but most won't get anywhere near....
my opinion of the later Sony units has dropped dramatically over the last few years, i was once a devout Sony user, but now i wouldn't use one if it were free....imo they are currently one of the worst sounding units available....
my advice to you would be;
1, sell on the Sony...
2, sound deaden your doors
3, buy either Clarion, Alpine, Pioneer or JVC, (again visit halfords and try out all the headunits they have on show that are in your price range, then buy online)...
4, hard wire new headunit direct from the battery using 10AWG cable fused at 20amps,(stick with original ign live), and earth headunit direct to chassis, again using 10AWG cable,(this will limit voltage drop).....
5, don't expect too much from your stereo....
a few interesting facts which may help you understand why your having problems...
1, an amplifiers output is measured playing a 0dB sine wave at a specific voltage,(either 12V or 14.4V), into a given impedence....
2, music can't be recorded at 0dB, its impossible....
3, as a speaker plays, heat is produced, as heat is produced impedence increases, efficiency also suffers....
4, volume or output scale is logarythmic, so 10watts isn't twice as loud as 5watts....half volume on your headunit isn't half power, majority of power will be in the top third....
so, just what is the approx output of your Sony when playing music???
say it's 4x 50watts max,
so actual power say 4x 14watts rms whilst playing a 0dB sine wave at 14.4V
actual continuous rms power whilst playing music at full tilt, probably around 7watts rms,(this figure will also be affected by impedence rise), so i doubt your 'over powering' your stock speakers...