The problem is with the wiring of your house. It is likely you have a loop system rather than a star system..
With a loop system, the main BT socket is connected to the next extension, and that extension to the other extension and so on to cover the entire house.
With a star system, each extension has a direct link to your main BT socket.
Because of the noise on your line, these disconnects are happening.
What you should do, is first find your main BT socket (you will know when you find it because it looks different from the other sockets). Remove the first two screws and then remove the face plate. By doing so, you have disconnected every other socket in your house and the main BT test socket will be live. After removing the faceplate, you will see another socket within. Plug your router in that and see if the connection drops. It is highly unlikely it will and you will also have a great connection as your router would be directly connected to the telephone exchange.
If this solves your problem, then ask AOL to call a BT engineer to your house. He will examine your existing house phone infrastructure and give you advice on what needs to be done. Ultimately, it will lead to either having every other BT socket disconnected for good, or he may be able to have one or two connected.
If your main BT socket is in a room you don't like, you can ask him to change it to the room you want..
Unfortunately, this is the price you have to pay to be on the BT ADSL MAX service. I think it's well worth it...