Do you have a spare POTS filter? If so try changing it and see if that makes a difference. I had a similar issue a couple of years ago and if I recall correctly changing the filter fixed the issue.
Before you buy any thing disconnect every thing just connect your modem ( no filter connected ) if the problem is still the same it is your line or your modem
When you take the phone off the hook you change the dynamics of the line . you are effectively putting a short circuit on your line when you lift the phone, that changes the characteristics of the line and should make any noise on the line less that is why it is working when the phone is off the hook.
I would say it is a line fault.
My adsl worked intermittenly until I got a phone. Then it worked for a few days and now its giving problems again. Will call them to check the line.
Thanks guys