UTMs disappear in URL, does the plugin still track?
Short answer is "absolutely yes"...
HandL UTM Grabber is a cookie based tracker. As soon as prospects hit your site with UTM data, they will be recorded in the COOKIES (client-side). And when ever they are needed (for conversion), they will be accessed from COOKIES. Hence, you don't need UTMs appended to every single URL in your website at all. You only want to append UTMs to the landing page URL to get recorded.
Having said that, if you are sending traffic outside of your site and want to pass UTMs to external domain. You can do so in two ways...
You can append UTMs globally to all the URLs in your site. See this article for more information Appending UTM Globally
You can append all the UTM parameters to a specific outbound link. Please see the article here Appending UTMs to the buttons using class name attributes (Selectively).