Add a settings page with custom form, then process the input. This bypasses the Settings API.

custom-wp-options-form-settings-page.php

<?php
/**
 *	Add a settings page with custom form, then process the input
 *	This bypasses the Settings API
 *
 *	@package Plugin_Name
 *	@author Ren Ventura
 */

// Exit if accessed directly
if ( ! defined( 'ABSPATH' ) ) {
	exit;
}

/**
 *	Add the menu page in WordPress
 */
add_action( 'admin_menu', 'myplugin_settings_page_callback' );
function myplugin_settings_page_callback() {
	add_menu_page( __( 'My Plugin Settings Page', 'textdomain' ), __( 'My Plugin Settings', 'textdomain' ), 'manage_options', 'my-plugin-settings', 'myplugin_render_settings_page' );
}

/**
 *	Render the page's content
 */
function myplugin_render_settings_page() { ?>

	<?php $options = get_option('my_plugin_options') ?>

	<div id="plugin-settings" class="wrap">

		<div id="icon-options-general" class="icon32"></div>
		<h1><?php esc_attr_e( 'My Plugin Settings', 'textdomain' ); ?></h1>

		<?php if ( isset( $_GET['settings-updated'] ) && $_GET['settings-updated'] == 'success' ) : ?>

			<div class="notice notice-success">
				<p><?php _e( 'Settings successfully updated!', 'textdomain' ); ?></p>
			</div>

		<?php endif; ?>

		<div id="poststuff">

			<div id="post-body" class="metabox-holder columns-2">

				<!-- main content -->
				<div id="post-body-content">

					<div class="meta-box-sortables ui-sortable">

						<form action="?save-plugin-settings=true" method="post" id="plugin-options-form">

							<div class="postbox">

								<div class="handlediv" title="<?php _e( 'Click to toggle', 'textdomain' ); ?>"><br></div>

								<h2 class="hndle"><?php _e( 'Do the settings!', 'textdomain' ); ?></h2>

								<div class="inside">
									
									<p class="">
										<label for="my_plugin_options[checkbox]"><?php _e( 'Setting Checkbox Label', 'textdomain' ); ?></label>
										<input type="checkbox" class="settings-checkbox" id="my_plugin_options[checkbox]" name="my_plugin_options[checkbox]" value="1" <?php checked( isset( $options['checkbox'] ) ? intval( $options['checkbox'] ) : '', 1 ); ?> />
									</p>

									<p class="">
										<label for="my_plugin_options[text]"><?php _e( 'Setting Text Field Label', 'textdomain' ); ?></label>
										<input type="input" class="settings-checkbox" id="my_plugin_options[text]" name="my_plugin_options[text]" value="<?php echo isset( $options['text'] ) ? esc_attr( $options['text'] ) : ''; ?>" />
									</p>

									<?php // Submit button at bottom of each box ?>
									<p class="save-settings"><input type="submit" class="button-primary" value="<?php _e( 'Save Settings', 'textdomain' ); ?>"></p>

								</div><!-- .inside -->

							</div><!-- .postbox -->

							<?php wp_nonce_field( 'plugin_settings_nonce', 'plugin_settings_nonce' ); ?>

						</form>

					</div><!-- .meta-box-sortables .ui-sortable -->

				</div><!-- post-body-content -->

				<!-- sidebar -->
				<div id="postbox-container-1" class="postbox-container">

					<div class="meta-box-sortables">

						<div id="plugin-settings-sidebar" class="postbox">

							<div class="handlediv" title="Click to toggle"><br></div>
							<!-- Toggle -->

							<h2 class="hndle"><?php _e( 'Settings Sidebar', 'textdomain' ); ?></h2>

							<div class="inside">

								<p><?php _e( 'Some information on the side.', 'textdomain' ); ?></p>

							</div><!-- .inside -->

						</div><!-- #plugin-settings-sidebar .postbox -->

					</div><!-- .meta-box-sortables -->

				</div><!-- #postbox-container-1 .postbox-container -->

			</div><!-- #post-body .metabox-holder .columns-2 -->

			<br class="clear">

		</div><!-- #poststuff -->

	</div> <!-- .wrap -->

<?php }

/**
 *	Process the input, and save options
 */
add_action( 'admin_init', 'myplugin_process_settings' );
function myplugin_process_settings() {

	// Bail if query arg is not set, or not correct
	if ( ! isset( $_GET['save-plugin-settings'] ) || $_GET['save-plugin-settings'] !== 'true' ) {
		return;
	}

	// Bail if no administrative privileges
	if ( ! current_user_can( 'manage_options' ) ) {
		return;
	}

	// Security check failed
	if ( ! isset( $_POST['plugin_settings_nonce'] ) || ! wp_verify_nonce( $_POST['plugin_settings_nonce'], 'plugin_settings_nonce' ) ) {
		wp_die( 'Permission denied', 'textdomain' );
	}

	$new = isset( $_POST['my_plugin_options'] ) ? $_POST['my_plugin_options'] : array();
	$sanitized = array();

	// Bail if no inputs are posted
	if ( ! $new ) {
		return;
	}

	// Sanitize checkbox option
	if ( isset( $new['checkbox'] ) ) {
		$sanitized['checkbox'] = intval( $new['checkbox'] );
	}

	// Sanitize text option
	if ( isset( $new['text'] ) ) {

		$sanitized['text'] = sanitize_text_field( $new['text'] );

		// Unset if empty
		if ( empty( $sanitized['text'] ) ) {
			unset( $sanitized['text'] );
		}
	}

	// Update option with sanitized array
	if ( empty( $sanitized ) ) {

		// Settings are empty, so delete from database
		delete_option( 'my_plugin_options' );

	} else {

		update_option( 'my_plugin_options', $sanitized );
	}

	// Redirect
	wp_redirect( add_query_arg( array(
		'page' => 'my-plugin-settings',
		'settings-updated' => 'success'
	), admin_url( 'admin.php' ) ) );

	exit;
}