I understand what you're saying about the dynamic range. I think that's something I could process from the ADC output.
No. The ADC goes from 0-1023 and that's all the "digital information" you get.
Any range/scaling has to be handled on the analog side. You can adjust that with amplification or by adjusting the ADC reference.
The 10-bit ADC gives you about 60dB of dynamic range. So for example, you could read from 40-100dB SPL (if it's properly calibrated to max-out the ADC at 100dB). That's not as much range as you wanted but it's enough should be enough for your purposes.
but what I found at Amazon was $800+. I was looking for < $50.
I don't know the exact price but like I said, this equipment is expensive. You can
rent one for less than $100 per day.
I live in an area that has a severe noise problem but the citizens cannot fight back against the offender without empirical data.
If your city has noise regulations, then the city ("code enforcement" department?)
might have a way to measure the noise. Otherwise, how can they enforce the regulations?
Of if the offender has "deep pockets" try contacting a lawyer. If there's money to be made the lawyer might help you collect evidence.