I don't think Time owns Time anymore, meaning didn't it get bought by some big conglomerate that uses the Time name?
Well anyway, YES, you should charge magazines for use of your photos!! Do the magazine and newspaper make money and benefit financially from using your work? Yes. They are not charities. Do their employees get paid to work there? yes they do, they may be contracted/freelance and not staff employees but they are paid/compensated and are not volunteers. Neither are you in this situation (it would be different if it's a nonprofit and you wanted to volunteer time or donate a service/product.)
If the newspaper was covering an event, that would be editorial use. If the magazine was covering a Chamber of Commerce event, that is editorial use too, but if the Chamber itself wants to use the photo for anything else, like their website, brochures, etc. that gets into commercial use. For either purpose you license usage and get paid/compensated for your photos. A photo credit is not compensation, it's just that, a photo credit.
Even if it's local and payment won't be a lot, still, you own your photos, they are your work. you own the copyright - you need to learn how to license usage of your photos. Any contract should be specific in terms of how the photo will be used, for how long, etc.
I think they count on the average person not knowing any of this and being thrilled that someone wants to use one of their pictures, so the company offers to use it free, for credit, or just plain ol' wants to use it.
Before you get any further with this, start getting informed and learning this stuff so you can start figuring out what to do and how to do it. I've gotten a lot of information thru
http://asmp.org or you might try PPA. It's fine to get ideas and ask for help here, also start reading up on how to do all this so you can be an informed professional photographer.