How do i export the HTML from text widgets?

The data in the options table is stored as serialized arrays. Use get_option() to get the data and unserialize them.

array_walk(
    get_option( 'widget_text' ),
    function( $d ){
        if ( ! empty( $d['title'] ) ) {
            printf( '<p>Title: %s<br>Text: %s</p>', $d['title'], htmlentities( $d['text'] ) );
        }
    }
);

If you need a complete plugin, use this. It’s a simple debug-plugin. Upload, activate and go to the dashboard. After activation there should be a Debug-Widget. If not, open the screen options and activate the Debug-Widget.

<?php
/*
Plugin Name: __WPSE__
Description: Testing plugincode
*/

add_action( 'plugins_loaded', function(){ new TestPlugin; } );

class TestPlugin {

    public function __construct(){

        add_action( 'wp_dashboard_setup', array( $this, 'add_dashboard_widget' ) );

    }

    public function add_dashboard_widget(){

        wp_add_dashboard_widget(
            'debug-widget',
            'Debug Widget',
            array( $this, 'output' ),
            $control_callback = null
        );

    }

    public function output(){

        echo '<div class="wrap">';

        array_walk(
            get_option( 'widget_text' ),
            function( $d ){
                if ( ! empty( $d['title'] ) )
                    printf( '<p>Title: %s<br>Text: %s</p>', $d['title'], htmlentities( $d['text'] ) );
            }
        );

        echo '</div>';

    }

}