The GWB can be done without a toll, but if you're shopping for a cheaper route to avoid the Verrazzano toll it's likely way out of your way.
Ideally the toll structure should incentivize through traffic to stay on highways and out of the most gridlocked streets in the country, and I'm saying this as a longtime car owner in Brooklyn who has driven through the Holland Tunnel hundreds of times. Manhattan streets are a tragedy of the commons in action.
But even with congestion pricing, assuming a same-day round-trip with a non-NY EZ-Pass, the combined cost of (Holland or Lincoln Tunnel toll + congestion pricing toll) is still a bit less than the cost of (Goethals or Bayonne Bridge toll + Verrazzano Bridge toll)... because the Verrazzano charges in both directions. So even congestion pricing doesn't fix those incentives :/
How so? The toll rate is identical between the Lincoln Tunnel, the Holland tunnel, the George Washington Bridge, the Bayonne Bridge, the Goethals Bridge, and the Outerbridge Crossing: https://www.panynj.gov/bridges-tunnels/en/tolls.html