I basically avoid the "Newport Zoo" because it's crowded on the land and on the water. When we go to the Boat Show we park on the street that runs along the water almost from the Newport Bridge to the downtown docks past the bridge leading out to Goat Island. I think it was Free.... but Restricted.
I would think that who ever is renting the moorings should have some info for you. If not maybe the city has some kind of Special Permit or something for "part time residents".
Although I infer you would prefer free parking, that is not an option anyplace close to the harbor unless you want to leave it on Goat Island. In Newport, you can leave a car on the street overnight but only if you have a sticker or like to take loooong walks.
I agree that parking in Newport RI is tough to find and expensive. I have stayed many times in Brenton Cove when in Newport and you can leave a car at the Fort Overnight for 50 cents or a dollar per nigt. Always room to park and you can dingy in to the dock there. Many times friends have come for visit and that is where I tell them to park. I can pick up with dingy or they can take launch into town and meet me. I will often dingy into Ann's Street Pier for a night out on the town.
PS: in thick fog (a Newport staple) good idea to follow shoreline into Brenton Cove so as not to exit Harbor accidentially.
You could also try down Wellington Ave. (That may be where the last poster meant...) I don't know the deal with multiday parking down there but it's worth a shot. There's also a dock down there you can use for your dink.