Dave has an HP 6700 printer which keeps disconnecting from his network. Clearly, since Dave has replaced the router and modem, the culprit is the printer. He even tried it on a wired connection, but it still dropped off the network. So that eliminates Wi-Fi congestion as being the problem.
It sounds like something has gone wrong with the network settings on the printer and since printer is several years old, it may be time to replace it. He could also try downloading the latest firmware and installing it. That can often solve issues like this. He should do that before he buys a new printer at least.