ITT: You're never finding that unicorn.
>Hired by a writing publication
>Wants me to do a branding, logo, website, and print
The /gd/ dream has finally come true.
>Wants me to build the website in WIX
I did a brand/website a couple years back. The logo won a Gold and a Judges Choice at an annual district advertising/design competition.
And the website? Webs. Webs dot fucking com. You think Wix is bad? Webs is Wix if Wix finally realized there was no value in trying to compete with Squarespace and succumbed to meth addiction and severe depression.
Try convincing them that Wordpress would be easy for them to update content through, especially for publishing, and that it would allow you significantly more freedom to design them the best possible website.
And if not, cut your losses and have fun with the logo.