Trying to make an amplifier to work
There are some issues here for you to address:





Keep in mind we can only see any picture you post here with 640 pixels wide maximum.  Lay out your schematic to not waste horizontal space.  The opamps could have been much closer to the output stage transistors, for example.  Remember that you are asking volunteers for a favor.

VCC_1 needs to be about &frac12; the supply voltage in this design.  That is your internal signal quiescent level.  Getting that right and low noise is important, but it is not shown.

Vcc needs to be large, like 24 V or so.  The TL07x opamps need several volts headroom at each end.  They are no good for 5 V operation, for example.  You generally want to use &plusmn;12 V (24 V total) at least with these opamps.

The deadband due to the double emitter follower output stage is a obvious source of distortion.  There is about 1.4 V range coming out of the opamps that isn't going to the output.  That could be a large fraction of the signal into small headphones.  It is no surprise this result in highly distorted and bad sound.

A quick hack to deal with the deadband is to include it in the feedback loop.  Instead of taking the feedback from the opamp output, take it from what you really want to control, which is the driver stage output.  With feedback around the deadband, the opamp will try to "jump over" the deadband quickly.  This won't be perfect, but good enough for basic operation.  I vaguely remember that the TL07x can slew about 8 V/&micro;s, so the little glitch at crossover should be small.

Considering you need the opamp to jump over the deadband as fast as possible, don't slow it down by using the feedback loop to add low pass filtering.  If you want low pass filtering, add it in front of the opamp.

If the opamp is unstable after including the output drivers within the feedback loop, allow it to have some gain instead of first trying to slow it down.  You currently have a gain of 3.8, so maybe no deliberate compenstation cap is required.  Try it and see.

The input caps are backwards.  Presumably the average level is ground at left, and half the supply at right.

22 &micro;F for the input caps is absurd.  With 18 k&Omega; in series, the high pass filter rolloff is 400 mHz.  That will take a long time to settle.  Just 1 &micro;F gives you 9 Hz rolloff with 18 k&Omega;.  That's much more suitable for audio.  Now you can use a ceramic and not have to deal with the polarity and lifetime issues of electrolytics.  You need to pick a ceramic that doesn't exhibit piezo effects, and that isn't too non-linear with voltage.  One thing that helps with the latter is to get caps rated for more voltage than you need.  The lower capacitance again helps with this.  For example, 1 &micro;F 50 V caps made with the right ceramic are more appropriate than the electrolytic caps you show.

220 &micro;F for the output caps may not be enough.  You didn't say what the impedance of the headphones are, but if they are 8 &Omega;, then the high pass filter rolloff is 90 Hz.  Conversely, to get 20 Hz, the headphones need to be 36 &Omega;.  At the least, check that this is all what you intended.