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I think that the printer's current Offline status has nothing to do with the new modem/router - if it is still configured to print through the USB cable directly from the computer.
(Check that the printer has not been mistakenly set to be a networked printer... it might doubtfully happen.)
Go to the printer's icon in
Control Panel - Devices and Printers.
Right-click on that icon, and see if a check-mark is next to 'Use Offline', if so, remove that check-mark. (Or, click on 'Use Online' if there is such a entry in the right-click menu and it does not have a check-mark next to it.)
If you want, after that, to try a full uninstall-restart-reinstall, let me know, and let me know whether your Windows 7 is 32-bit, or 64-bit. Thanks.