It really isn't very clear what you're trying to do. If the people you're staying with have a *MODEM* connected to the internet, then all other computers will get their internet access via a *ROUTER* used in conjunction with their *MODEM*.
You cannot connect your Belkin combined "Router & ADSL Modem" to ethernet sockets simply by inserting an RJ11 (phone) to RJ45 (ethernet) adapter into the circuit.
>>> Is there a way around this or should I just buy a new modem.
I'm not sure that you aren't confusing a MODEM with a ROUTER. A Modem is the device that connects to the internet service, either DSL over the phone line or Cable. A Router allows multiple computers to connect simultaneously to the internet.
The hardware you need depends on what the ethernet sockets in the rooms are actually connecting to. If they already connect to a Router, then all you need to do is to plug your computer directly into the socket for internet access.