Since you have a permit from another state you are eligible to apply to any town (usually Chief of Police) under RIGL 11-47-11, and they are a Shall Issue authority.
Most departments have applications, only Warren, Barrington, Central Falls, and New Shoreham (Block Island) doesn't have applications. However, the New Shoreham Chief has said that if someone wanted to apply, he would come up with a process.. The Chief even said that there is nothing that requires him to have a "written application", which I agree with him on, but it makes it a little easier if he developed a policy. Most other towns have issued permits, but many try to direct you to the AG. However, if you let them know that you intend to apply under 11-47-11 and that they are required by law to accept and process these applications (Archer v. MacGarry) they will give you the application. Sometimes you have to request the application under the State's Open Records law.
Unless you have a very good reason to carry a firearm you would like to share with the AG, I recommend that you apply to a town in RI, preferably to a town that you visit frequently. That doesn't necessarily mean that you will get your permit since Chiefs are notorious to violate our rights when it comes to CCW.
It was Shitehouse (I mean Sheldon Whitehouse) while he was the Attorney General who instructed the Chief's to not accept applications, and unfortunately he is now one of RI Senators in Washington DC.... When the current AG was asked to send out a letter to the Chiefs asking that they follow the law, the response was that they have no jurisdiction over the Chiefs...