Do you know the model number for your modem? ADSL and ADSL2 are pretty established technologies. Your modem would have to be a decade old to not support ADSL2, and even regular ADSL supports speeds well over 700 kbps. Unless your modem is 10-20 years old, I really doubt the problem is because your modem is old (unless it is malfunctioning). If your modem really is old, you can try replacing it. But I'd get the replacement from someplace you can return it for a refund if it doesn't fix your problem.
Do you have phone service on this line as well? Make sure you install the DSL filters on all phone jacks with phones plugged in. DSL and phone service are carried on the same wire. The phone service is carried at the lower frequencies, while DSL is carried at the higher frequencies. A phone can add high frequency noise to the line which can interfere with DSL. The filters remove that noise.
If the problem is the phone line, you're going to be in for a long fight. Verizon will do everything it can to avoid admitting the problem may be their line, so they won't have to pay to fix it. They will blame your modem, they will blame your router, they will blame your computer, they will blame the wiring in your house. Not because a technical analysis says those components are at fault, but because those are components the customer has to pay to fix, not them.
I fought with them for 2 years for a line problem that appeared every time it rained. Every time I'd open a ticket, they'd test the line a couple days later (when it wasn't raining), say their line was fine, and close the ticket. I finally nailed them when getting a T1 line installed. It happened to be raining during the install day, and the company installing it told me they were talking to Verizon about connectivity problems. I asked them if they had a Verizon tech on the phone at that moment. They did, so I immediately told them to request a line quality test. That was when Verizon finally admitted the line was faulty and sent out a repairman. (Turned out a bird had built a nest on one of their poles, and picked apart the insulation on some of the phone lines. Every time it rained, the water would contact the copper and cause shorts/static.)
If you're convinced the problem is a faulty Verizon line (e.g. you also experience static on phone calls), it may be easier just to get a new line than to try to get them to fix the original one. Most homes have two phone lines coming in (4 wires = 2 twisted pairs = 2 phone lines). I'm assuming you already have phone + DSL service on one line. Request another DSL line (a dry loop - DSL without phone service). This will set up DSL service on your second line. Hopefully it does not have the line quality issues of your first line. If the second line works, you can just cancel the DSL service on the first line. (After testing, you'll have to cancel service on one of the lines. So be mindful of the length of the contract and any cancellation fees. You may be better off keeping the second DSL line and canceling the original DSL service.)