Potentially, yes, but that will depend on the agreement made between the provider and Apple as to what content will be made available as part of the Apple News+ service. I believe that you don't need a subscription to WSJ to access it with News+ ... so having a standalone WSJ subscription would be redundant.
Using WSJ as an example, with your trial, look to see if you are provided with the full content of the paper journal ... or, at least, the parts that you would be interested in. If all you get are selected stories without the ability to gain access to the full content, well then, you have your answer. I would do the same for any other newspapers or magazines you currently subscribe to.
Note that the trial provides full access to the News+ service. It doesn't include anything else once the trial period is over and you decide to subscribe to it. Basically, you get what you see.