If he is a sex offender, than more likely than not he must register with the local sheriff and keep them up to date of any address changes. You can check his status with the sheriff and if he's not up to date they may arrest him. Beyond that, you're always free to find a new place to live.
Those guys are right on. Also, if family with kids visit, your landlord is not breaking the law being next door, unless he is on probation and his probation calls for him to be a certain distance from kids. I would certainly tell the family about the neighbor, though, before they visit!
As far as him living next door and having your address goes, that could have been a technical violation of the registration statute, if he had moved and not updated. Will they arrest and prosecute it if you report it? Unkonwn. Depends how zealous they are. Cops and prosecutors love to beat up on sex offenders. It's a very popular thing to do. However, that would be such a ticky tack violation of the law and it sounds like he has already corrected it... so it would seem to be a silly prosecution.
I guess the big question is, how much do you hate your landlord? If he just creeps you out you should just move and move on with your life.