I just got done with a dying SB5120 that performed similar to what you describe. To get self-diagnostics from the modem just enter: »192.168.100.1/
and you can pull at least 5 pages of good stuff. If you get an unstable connection, that's a pretty good indicator of a bad modem. If you can hook up with the modem, you'll get signal levels up and down, SNR, and a log of occurrences that if a lot are code 3 timeouts could be indicative of modem probs. But your best bet is to join up--it's free--and either study the FAQs re modem problems or just IM your modem MAC address to CoxTECH1 or CoxTOC1 and they can advise based on what they can read.
If you set up a service call, tell them you want a SB6120. It's now on the supported list and is the coming thing--and looks real Space Odyssey compared with the twinky SciAtlanta I got stuck with. No matter what, you need to run with a PC plugged direct into the cable modem for a few days so you eliminate or confirm the router as a trouble source. Have fun!