I kicked out OE and put up Thunderbird (been running that on my laptop already and had planned it already for awhile for my desktop), uninstalled Avast and reinstalled it and after that it worked. But it still doesn't work in OE. I've tested it there afterwards, and it still refuses. So there must be some kind of problem in OE. Can't bother with it anymore, though.
Thanks alanrf, for the time you put in this. Much appreciated.