Problem solved! Before buying a new modem, I decided to try one more thing. The network cable FairPoint sent me has a Y on the end, so it can be connected either into a network card or usb port. Their instructions said to use the network card rather than the usb, if one was available, so that is what I did. I know the net cards on both the desktop and laptop computers work, because I frequently use them to transfer files back and forth.
I tried connecting the modem through the usb port and then the email worked and the web pages stopped hanging. I reconnected it through the RJ45 and was back to the same problems. Although I had done it before, I tried a different network cable that I knew was good, and again had problems.
Now I think there must be some incompatibility between the modem and both of my network cards. Both computers are running Linux, so I don't know if there is some issue with the drivers that they will work with my router but not the DSL modem. Anyway, everything looks good when I connect through the usb, so that is what I will do.
Thanks again to everyone for helping me to troubleshoot this. I hope this discussion may be able to help someone else with similar problems.