How do I get paragraph fields text data without the  tags?
Does your field contain other &lt;p&gt; tags you want to preserve? Or just this one in the beginning and end? If you don't need to preserve other &lt;p&gt; tags, you could simply use Twig's |striptags filter.

&lt;p class='my-custom-class'&gt;{{ my_var|striptags('&lt;p&gt;') }}&lt;/p&gt;


Or as @Hudri suggested in the comments under your question you may change the field type to something other than formatted text. Or maybe try changing the text format to plain.