I had to get a waiver. I do not have a full sleeve like you, but on my right arm I have more than the hand can cover exposed (in a short sleeve shirt). I have a half sleeve that extends about 1" below the elbow. My other arm just has a nautical star on the elbow.
I read in a forum the other day, in regards to officers w/ exposed tattoos, a sailor stationed down in Pearl has seen a countless number of Mustangs that look like they spent every night of their enlisted days in tattoo parlors. On the Navy OCS Forums, there is a thread about tattoos. One of the responses was that it wasn't a problem, the officer knew two other officers w/ full sleeves on both arms. I was however told not to expect to ever make it past 0-3E with exposed tattoos. Not sure if it was true or just someone hating because I know USAF and not USN, but in the USAF Os w/ tattoos is highly frowned upon. It would seem that in the USN it isn't..........who knows.