Conservatives: what sources would you consider? (details)

As a social liberal and fiscal moderate, I often find myself involved in debates with politically and socially conservative types of people, both on Fluther and elsewhere. Often, in these debates, I like to bring up facts and figures to support the points I’m trying to make.

Now let’s assume that the fact I’d like to reference is in direct contradiction to a conservative point (hence the debate). As such, it won’t be very likely to be reported on FOX or any other typically conservative-leaning site, but it is a fact nonetheless. Yet if I link to a typically liberal or even neutral site, I’m often told that it’s just media spin and probably a lie. As such, it’s written off as unimportant to the debate at hand, since the existence of the fact does not bolster the conservative opinion. The debate, instead of being a useful dialog of fact and information, breaks down.

Conservatives, with all due respect, how can I best point out to you a fact that you might not be aware of, if it is not on a conservative-leaning site? What sources and sites (outside of the Conservative echo chamber) would you be most likely to believe? Can we agree that it is possible for there to be a fact out there that might, if learned to be true, change your opinion on some matter? Or is there some debate procedure, beside the introduction of facts and information, that is better suited to the way that you think, that might work better (such as a good story, appeal to faith, quotes from noted Conservatives, etc.)?

In short, Conservatives, what is the best way to turn our debates from unproductive shouting matches into a productive exchange of ideas, information, and thought? What would make you most willing to listen to a contrary point, even if you choose, upon hearing, to not believe it?

I’m willing to meet you halfway, if you’ll tell me how. Thank you.