WSJ is definitely fiscally conservative, and otherwise fairly moderate.
Economist is actually quite liberal leaning.
It's possible that your personal bias is very liberal so normal "conservative" positions seem far right. To be fair, news as a whole tends to lean slightly liberal.
http://guides.lib.umich.edu/c.php?g=637508&p=4462444 is a decent source for news bias alignment.
On to the point of whether publications like the Economist are left or not. This may be a matter of whether you consider neoliberal policies to be conservative. I personally believe that the Economist's position on privatizing industries, selling national assets, minimizing worker protections in the form of minimum wage laws, etc. to all be extremely conservative.
I happen to be more sympathetic to opening borders to immigration, which the Economist supports. And the Economist is very much anti-death penalty, and, as others have pointed out, they are socially liberal (in the US sense) in a number of other ways. But at the end of the day this is a magazine called The Economist, and their economic views are mostly in direct opposition to mine.
But I'll still read it, because I think it's a mostly fact-based publication.
Economist describe themselves as "extreme centre" and have supported both right and left at different times. So they're likely to remain hard to precisely pin down. :)
The reason for that is the bias of any given position is very difficult to calculate and political identities are heavily nuanced across the entire spectrum. In addition, most people don't subscribe to well fixed ideological positions on most issues - and the line of being moderate shifts over time.
But yes, as I said earlier, most news is a bit liberal leaning so news isn't perfectly aligned and in general there are much more liberal news sources then conservative news sources.
In general your best bet is to read a lot of news sources, understand that all news has some bias (towards increasing readership, as well as a slight liberal lean), and do your best to kind of meet in the middle. Be aware of general logical fallacies and try to correct for them as best you can, and always question everything.
I've started to think the idea of left and right wing are grossly over-simplistic. They might have made sense in the days of mass media where there was no room for personal differences but now you can find popular Youtubers who support a mixture of traditionally left and right ideas.
For example, they advocate protecting marijuana legalizing states at the federal level. (https://www.nationalreview.com/2018/06/marijuana-policy-shou...).
A lot of their positions (taxation, free education, government spending in infrastructure, universal healthcare, etc.) would be classified as left-leaning or left.
Additionally, I've seen that leftist news sources aren't by any means behind rightist ones in terms of fake news. They just focus on different sets of fake news.
Maybe the fact that you find rightist news more biased is just a result of your own bias. You overlook the bias in the publications that are aligned with your beliefs.