I have a custom taxonomy assigned to two post types. How do I create URLs for different term archives for each post type?

Question

I think your issue is with the rewrite portion when you registered the custom taxonomy with posts.

'rewrite' => array('slug' => 'some-slug', 'with_front' => false)

You can replace some-slug with anything you would like. It might be possible to replace it with article/event-category to get the results you want.

Setting with_front to false removes blog as a prefix. If you have changed the blog prefix to article. Then you should be able to set with_front to true and replace some-slug with `event-category

Here is a link with more info on registering custom taxonomies
https://clarknikdelpowell.com/blog/the-right-way-to-do-wordpress-custom-taxonomy-rewrites/

EDIT:
Can you do the following completely untested, created example code from link below:

register_post_type(
    'post',
    array(
        'taxonomies' => array( 'event-category' )
    ),
    array(
        'rewrite' => array('slug' => 'some-slug', 'with_front' => false)
        // other arguments may be needed from other registration of taxonomy
    )
);

Reference:
Can multiple custom post types share a custom taxonomy?