Your fill options in Southern RI are the Newport Dive Center mentioned above, Pier Dive in Narragansett, or if you head a bit north away from the coast there is Anderson's Ski and Scuba in North Kingstown or Giant Stride in Warwick. All of them except Anderson's also do Nitrox. Some of the shops have better service than others but for something like an air fill it doesn't make much difference so I'd just use the closest one.
Personally, although there is a lot to do there, I'd avoid staying in Newport just because it is generally going to be pricier and it will likely be a zoo on labor day. Instead you could just drive over there if you want to visit Newport and stay elsewhere. Narragansett or South Kingstown (Wakefield and Matunuck in particular which are both part of SK) would be places to look. That will put you on or near the south coast within easy driving distance of Newport, Jamestown or Westerly. Nice central location, pretty coastline, and not quite as crazy or pricey as Newport.
For shore diving, Jamestown probably has the best options. Other than Ft Wetherill (which is a nice dive but often busy and can be a bit murky) there is Beavertail Point which is a great dive (Parking lots 2 or 3 are probably the best options though 4 can be good too.... 1 has terrible shore access and isn't really worth the effort). Beavertail require relatively calm seas and has a fairly difficult entry exit on the rocks so I'd only recommend it if you are in good physical condition and are comfortable climbing slippery rocks in scuba gear.
Newport has a couple of shore sites, Kings Beach is probably the most popular one. That's a pretty easy shallow dive.
Narragansett also has a some nice sites. State Pier #5 (aka Monahan's) at the end of the sea wall in Narragansett is very easy shallow site, though it is exposed so surge can be an issue. It does however have pretty repetitive terrain so you can get bored pretty quickly, and in general I find it to be a better night dive than daylight dive. Bass Rock Rd can also be a nice shallow dive when the surf isn't too big although the hike down to the water is a bit of ankle buster. At Bass Rock if you go south from the entry point you get big boulder islands across a white sand bottom. If you go north you get the wall along the big rock face you'll see when you get there (you'll see what I mean if you Google Earth it).
As for boat dives. The famous one is the U-853, which I am sure you've probably already run across in your searches. There are dozens of other nice ones as well, but you'll probably be limited by what the charters are going to.
