Take 95 over the Throggs Neck Bridge, the Whitestone is deeper into Queens and is a construction nightmare. It seems like they are always working on the bridge on the queens side. New York now sends you toll bills in the mail unless you have e-z pass. I'm not sure about toll cost, I have e-z pass so I don't pay attention to the cost, it auto replenishes.
There is a toll in New Rochelle, other than the bridge I don't know of anymore tolls. Throggs Neck is $9.50, that's the car rate. I would go after rush hous before construction. Beaches are a nuisance but that's over now. Saturday or Sunday morning are your best bet traffic wise, consider that because traffic coming off the island on Sunday can be a bear. DO NOT GET ON PARKWAYS WITH THE TRAILER! The fines will be hefty!