Have you considered just paying? People on this site often complain about freeloaders wanting software or support without offering anything in return, yet they also prefer to bypass paywalls on content.
I have seriously considered paying for the NY Times and that's the only site I've considered. I still haven't seen any reason to pay, yet. Seems like I can always find something somewhere else to read, and nothing that's exclusive to The Times has really drawn me in that much, at least not yet. I am aware of the freeloaders thing, and I end up feeling like one sometimes.
The problem that I face is that I don't get $20/mo of value out of any one publication (ok, I guess NYT is $17 per 4 weeks, which is a bizarre billing cycle). I certainly get (more than) $20/mo of value out of "online news" as a whole, but it's spread out, and any given site would probably only get a few dollars a month (and many sub-$1) if allocated what I'd consider fairly. But pretty much no news website will let me subscribe for that little, or will give me a cheap plan that limits me to, say, 10 or 12 articles per month (which would still be overkill for many of the paid sites I'll occasionally click through to).
If I were to subscribe to every site where I've bypassed a paywall even once in the last few months (of have just given up without reading), I'd easily be spending over $300 per month. I don't feel like my consumption warrants that price.
Newspapers have, to the best of my knowledge, always billed in week increments. I agree it's a bit strange, but I believe it's driven by a need to sell ads by the week and esp. on Sunday.