Use <label> tag before <input> like this.

<label>Email Adress:
    <input type="text">
</label>

This is a simple method to do this. You can simply omit the use of the submit button.

Here is the modified code of yours.

<div id="psform"> 
    <?php $search_text = "Search"; ?> 
    <form method="get" id="searchform" action="<?php echo home_url(); ?>"> 
        <label>Email Adress:
            <input type="text" value="<?php echo $search_text; ?>" name="s" /> 
        </label>
    </form>
</div>

Edit:

If you want to add a title to the search form, you can use any heading tag. If you need any other help, please don't hesitate to ask.

Use <label> tag before <input> like this.

<label>Email Adress:
    <input type="text">
</label>

This is a simple method to do this. You can simply omit the use of the submit button if you don't need it to show.

Here is the modified code of yours.

<div id="psform"> 
    <?php $search_text = "Search"; ?> 
    <form method="get" id="searchform" action="<?php echo home_url(); ?>"> 
        <label>Email Adress:
            <input type="text" value="<?php echo $search_text; ?>" name="s" /> 
        </label>
    </form>
</div>

Edit:

If you want to add a title to the search form, you can use any heading tag. If you need any other help, please don't hesitate to ask.