Hello and welcome to the forums.
I am not sure if you have tried calling the BT Residential Broadband Help desk (for the little use they might be) but they would have asked you to try your modem at the master telephone socket.
This is the first telephone socket connected to the line where it enters your property. From your post it sounds like this master socket might be the one located downstairs into which you have connected an extension lead?
The purpose of this is that any broadband help desk will ask you to try and connect the modem at the master socket with no extension leads or hardwired extension sockets connected. This is to help determine if the problems are present on the outside line or are caused by internal factors such as poor/faulty wiring, long cable runs and electrical/radio interference.
The Speedtouch 330 is a USB only modem, so you cannot connect it to your computer using an ethernet cable/port.
It is odd that you loose access to the internet yet your modem claims to remain connected. When this next happens, check the ADSL light on the modem and see it if stays steady or or if it goes out/starts flashing which would indicate the broadband has lost connection.
As much of a pain as it is, you may want to identify the master phone socket in your property and try the modem there. I appreciate this might involve relocating the computer temporarily.
If you have an NTE5 style master socket (see link below), then try connecting your modem to the test socket by removing the front panel.
Also consider trying a different microfilter and modem cable as sometimes these can become faulty, although this is unlikely since the problems only began when you moved home.
Sorry this is rather long and complicated, but hopefully these website pages will help you understand a little about what I am trying to suggest
ADSL Line + Connection ProblemsImprove your ADSL connection speed (try at master)