lol Pete,
There are lots of people inside the IT world who haven't a clue about wireless, so don't beat yourself up.
I'd be surprised if changing the router did much for the wireless dispersion around your home. That is governed more by things that block signal i.e. foil lined insulation, stone and brick walls, steel lintels, bodies of water, the aggregate effect of walls and floors over distance, and the position of the router. If it is shoved up in a corner because it looks neat then that will mess up the dispersion.
The other thing is sources of competition. A typical 54g router works in the same frequency range as cordless telephones, wireless baby monitors, bluetooth devices, 2.4GHz TV senders, etc. In fact most wireless devices try to share the same frequencies. It's crowded. Then there's your microwave. Heat up a bowl of soup or defrost a loaf and it will dump interference all over channels 2 to 7. Don't forget as well you have neighbours using these devices and their own routers too.
N routers are less prone to interference because they use the 5GHz frequency range as well as 2.4GHz. However, unless your TV uses 5GHz too then there's no benefit.
Your typical Help Desk jockey probably won't have a clue about this, or if they do they'll be told not to discuss it. Sending out a new router makes it look like they're doing something and most customers would rather try that than be told it's because of the house or next doors wireless door bell.
Let us know how you get on, but in the meantime try moving your router about.