EZ Pass website says debts can be settled in cash at their service centers. So yes you can settle in cash, but it is likely a massive pain to do as you suggest. Their site also says you can prepay your tolls in cash, to have account in credit, using the same service centers. I suspect almost no one does.
"Cash or Credit/Debit cards are accepted at any of our Customer Service Centers."
… so at the time of the example, I would have needed to travel ~3,000 mi, then.
And… you have to sign up, get a transponder, keep said transponder in your car and allow them to track your movements with it. (Which, admittedly, they can probably already do by reading license plates. Might as well just broadcast it on radio?)
"Cash or Credit/Debit cards are accepted at any of our Customer Service Centers."
> https://www.ezdrivema.com/ezpassma
> https://www.ezdrivema.com/ezpassservicecenters