If you are certain the entire circuit is assembled correctly, then try this:
See the two 470kΩ resistors in red, in the schematic EL34 posted? Rather than have 1 resistor to ground and the other to the grid of V2/V4, place the 2 resistors in series after the 0.022uF cap (so 0.022uF cap, then 470kΩ, then 470kΩ then ground). Connect the junction of the 2 resistors to V2/V4's grid.
Now you have a voltage divider that's almost 1MΩ total load to the previous stage, and cuts signal strength by half.
... the overdrive is apparent at pin 7 V2 with massive gain there.
I'm wondering if I should change the cathode resistor on V2 - A at pin 3 (820R) to a larger value to reduce the gain.
Gain causes distortion so:
... V2 pin 2 and V4 pin 2 sound "overdriven" when turned up ... there is high gain and overloaded distortion at V2 and V4 pin 7 will little or no clean there
"Gain" is not the only thing that causes distortion.
If you added a diode to ground at any preamp tube plate, you'd get almost no clean output; it would all be half-wave rectified and sound very heavily distorted.
Similarly, if you had an 82Ω resistor at the cathode of V2/V4, bias would be insufficient to accept almost any signal before saturation, and would lead to distortion. In this case, the trouble really is the small range of allowable input signal. If you have a 100kΩ plate load at V2/V4, but accidentally used an 82kΩ or 820kΩ resistor, V2/V4 would distort but now it would be because they are biased to cutoff so strongly the signal gets clipped on the other side of the waveform.
There are other things which could be wrong, but hopefully you get the idea from the above.