Can’t seem to do combined query AND sort?

I had this same problem until I realized you don’t actually have to query by the event start date, only the end date.

As long as the event’s end date has not past it doesn’t matter whether the start date is in the future or not. All you have to do is grab all the dates that have not ended yet then you can order them by their start date.

This should work for you:

<?php
    $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
    $today = date('Y-m-d', strtotime('-6 hours'));
    $upcomingevents = new WP_Query(array(
        'post_type' => 'events', 
        'posts_per_page' => 6,  
        'paged' => $paged,
        'meta_query'=>array(
            array(
                'key' => 'evend_date',
                'value' => $today,
                'compare' => '>=',
                'type' => 'CHAR'
                )
             ),
        'meta_key' => 'evstart_date',
        'orderby' => 'meta_value',
        'order' => 'ASC'
    )); 
?>