It sounds like there's a timeout setting in your PPPoE client that is too short for Verizon's servers. I have no idea how to tweak this in MacOS, but I don't think that's a big problem-I have a hardware solution. (tooki's rolling his eyes about now, and thinking "of course he's going to roll out a hardware solution...")
Go to OfficeDepot, OfficeMax, CompUSA, BestBuy, or whathaveyou, and purchase an inexpensive gateway router. I haven't seen mention of one that doesn't have a good PPPoE client built in, so just about any will do. I recently got a Siemens 2604 for less than $30 (I just got the $20 rebate check last week-the thing cost just over $11 after rebate!-it even has a network print server built in!) Setting up one of these little boxes is as simple as pluggin it in, connecting a network cable to it, and entering its configuration page address in your browser.
I don't know what Verizon's terms of service are, but as you mention only one computer, you shouldn't have to worry about violating them. You are simply going to use this router to connect to the service you're paying for. It doesn't hurt that gateway routers also perform Network Address Translation, a sort of smokescreen to hide your computer from the outside world. NAT isn't really a firewall, but it's a good fireproof curtain. Anyway, if a gateway router's PPPoE client doesn't get through, Verizon has some 'splainin' to do.