So Verizon sent me a new modem after I told them it was their network. As I unplugged my Netgear, the red light of death was glowing hard. Plugged in a new D-Link ASDL router and...so far so good.
We'll see, Verizon, we'll see.
Update - Nope. Same connection problems, same drop-offs with new modem. As I suspected.