As I was gently trying to steer you towards, and as Alex said outright, the problem is those 100k resistors.
You say you're not a pro, so you hopefully won't mind if I suggest you read up about thermal noise:
https://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noiseBasically, the higher the value of resistance, the more Johnson (thermal) noise it makes. There are online calculators that save you having to do the maths. For example:
http://www.sengpielaudio.com/calculator-noise.htmA 100k resistor at room temperature makes about 5.7uV of noise - that's about -103dBu. Assumed bandwidth of 20kHz.
By contrast, a 10k resistor makes 10dB less.
Now, as Alex pointed out, swapping op-amps might only make a small difference - he suggested 2dB, which if realised in practice is hardly noticeable in reality. Whereas changing all those 100k resistors for 10k resistors will make a lot more difference. Resistors are cheaper than op-amps, which I hope is good news!
That was the point I was trying to make earlier - there are many DIs using the same TL06x op-amps that don't have noise problems, therefore the problem with the DI20 is not the op-amp. I hoped that logic was simple to follow...
So what would I change?
Those 100k resistors need to go. I'd make them 10k or less. In fact, I'd make the first one (R13B) 10k, and R13D 1k. I'd do the same for R13A and C. But of course, R13 is a resistor pack, so that might not be physically easy...
Also, R19 and R21 also need to be reduced from 100k to 10k. If you were feeling especially diligent, you might also increase C6 and C9 by a factor of 10.
Having done that, the input of IC1A is now a bit exposed, so I'd add protection diodes from the junction of R13B and R13D to ground and the supply rail.
With this done, the DI will present an input impedance to unbalanced sources of 21k when the pad switch is set to 0dB, but it will fall to ~10k when the pad is engaged (if you want to use the pad, you'll obviously have to reduce R9 and R7 by a factor of 10, naturally). But if you're feeding this from an FX or preamp pedal, then Zin of the DI is not an issue.
I'd also increase R2A and R2B to 1k or more, as the TL064 struggles to differentially drive a 1k2 mic pre-amp. A lot of DI boxes make this mistake, resulting in more distortion than is necessary.
Doing this will reduce the noise considerably more than any op-amp swap could ever do. The noise contribution from the 100k resistors is swamping the op-amp noise - even a relatively unsophisticated op-amp like the TL064.
Frustrated with the "schoolboy errors" in many commercial DIs, I make my own. It's really not difficult. I get the impression that in big companies, the DI design jobs get given to the new engineers while the experienced guys work on the high-value gear. That's why small companies like Orchid, who I mentioned earlier, make excellent DIs, as like me, they've actually thought it through...
Does that help?