How can I simplify this JQuery validation process?
I see you removed the calls to $('#loadingDiv').show(); from the many conditional blocks and moved that before the call to the AJAX call - I was going to suggest that.
I see the ecmascript-6 features like the Class syntax are used, so other ES-6 features could be used - like the object initializer shorthand notation - i.e. on this line:

 $.post('process/editUser.php', {uobj:uobj}, function(data)


The second argument can be simplified:
$.post('process/editUser.php', {uobj}, function(data)

There is no need to use the else keyword since all previous conditional blocks have a return statement.
The callback to the AJAX call has a variable modal:

var modal = &quot;#addUserModal&quot;;


The variable is only used one time so the value can be used in the call to dataCheck and the variable eliminated. Additionally, unless there is a good reason like needing a global variable, don't use var. Use const and then if you determine re-assignment is necessary use let.
Instead of using .indexOf() to determine if a string does or does not contain a substring, String.prototype.includes() could be used unless browser compatibility is an issue.
I know that the PHP PSRs recommend having opening braces on a new line but this is less common in JavaScript.