How Do I Get My Post Tags to Alphabetize?

You could use the filter term_links-post_tag to sort tags while ignoring their case:

add_filter( "term_links-post_tag", 'themeslug_sort_tags_alphabetically' );

function themeslug_sort_tags_alphabetically( $tags ){
    natcasesort( $tags );
    return $tags;
}

This code is to be placed in the functions.php of the child theme.

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