Thanks for the suggestion but I did reconfigure the modem via Yast several times. Even powered off, disconnected the modem, powered on, removed modem and provider via Yast, powered off again and back on, and the Kinternet icon on the root and other user's desktop showed with an exclamation mark saying 'modem not detected' or something, but no icon on my own user desktop and the same message 'already running' if I tried to connect. Then powered off again, re-connected the modem, powered back on, configured modem and provider in Yast, and the Kinternet icon on the root and other user's desktop showed normal and connected normally. But my own desktop kicker didn't show the icon, and when I tried adding it I got a different icon (the gear figure) - and either clicking that or running Kinternet from the Start button, both still give me "Kinternet is already running".
The problem is only with the one user (me) - root and the other user have no problem!
Eight months ago when this first happened and I was still on SuSE installation support, the support persons there couldn't figure it out. But surely there's some way to convince the KDE gremlins inside that Kinternet is NOT still running for one user!