The Crowment Comments plugin swaps the built-in WordPress comments for the Crowment widget, so you don't have to touch your theme.
Sign in to Crowment, create a comment widget for your site's domain and copy its app key from Install & Embed.
Download crowment-comments.zip. In WordPress, go to Plugins → Add New → Upload Plugin, choose the zip, select Install Now, then Activate.
Go to Settings → Crowment Comments, paste your app key and save.
Your old WordPress comments stay in your database; they are hidden while the plugin is active. To update the plugin, upload the new zip the same way and confirm replacing the current version.