I'm like you - I prefer to fix before replace just to try and find what is wrong, but if Norton has detected bad sectors then the system files that you need for normal connectivity could be on those bad sectors and that your computer finally is infection free.
As you are able to get a wired connection, you can download your Network drivers and save them onto a Flash Drive as the ISOs don't include any drivers and then you can reinstate connectivity after you have swapped out the HDD - but if you can boot up into Safe Mode either by tapping F8 (probably) or through msconfig, select Safe Mode with Command Prompt and enter chkdsk without any parameters to see if that will run in read-only mode, as any 3rd party conflicting programs won't be loaded - whether that is µTorrent, BitComet or whatever else you have downloaded and without a timestamp on this, there's no knowing which program that could be.
That will report any corrupt files/bad sectors without repairing anything.
If that still doesn't run, HDDScan 3.3 is a program (if you are able to download) that you can run S.M.A.R.T. and other tests on and that will show if there's anything wrong with the HDD - but once a HDD develops bad sectors then it's only a matter of time before it fails completely and when you are fortunate enough to have forewarning of that - you have time to back up for a replacement.
http://hddscan.com/A chkdsk /r doesn't repair bad sectors - it just moves what data it can to the good areas of the disk, repairing any system files and marks the bad sectors so nothing else is written to them and if those bad sectors have non system data on them - then that data can be lost.