If you have to reboot your Verizon router all the time putting in another router after the Verizon router is NOT going to fix or tell you anything other then you still have to reboot your Verizon router.
The DSl modem / router is a modem with a phone connection built in for the DSL modem. If you want to try anything go buy a DSL MODEM/ Router from best buy and configure that with whatever account settings Verizon uses out therefor your DSL account into the new router. Usually its PPPOE if the problem still happens its an issue with the line from your modem to the Demarc of the house or from the Demarc on the side of the house out to Verizon in the street or even further into the central office.
Most DSL modems you can log into and see whats going on such as if it has Sync meaning its showing connected to the Verizon network. If you log into your Verizon router usually »
192.168.1.1 username= admin password= password or password1 unless its been changed you can go under diagnostics somewhere and look when the problem is happening.
I would suggest that first before spending any money on another DSL modem/ router. if its not showing sync then have Verizon come out and look at your lines or at your equipment.