Before getting heavily involved with verizon do a simple test to discover where the problem originates. If you have access to the box where the the telco line enters your house connect a dual line dsl splitter like this one
http://www.excelsus-tech.com/index.c...ail&id=1749728 to the rj-11 jack on the outside box. Then connect your phone and your dsl modem at the same time and see what happens. If you continue to have problems it's your local phone company's fault and not the fault of your house wiring.
If the above configuration works without problems then your house wiring is at fault. Depending on the state you reside you might be responsible for that wiring. In California the homeowner is responsible for house wiring and not the telco, unless you sign up for the telco's insurance plan to fix house wiring. If you are in Calif. and you never signed up for the local telcos insurance plan then you are responsible. Luckily there are a ton of phone guys out their doing this on their own.
Also if you comfortable with rewiring your own stuff you can get info from different sites like this one.
http://www.homephonewiring.com/index.html