Instead of sending data in querystrings, which there is a limit to, as you have learned, add the 'Data:' attribute to .ajax call:
$.ajax({
type: "POST",
url: "ajax_functions.aspx/WebMethod1",
data: "{ action: 'insert_questions&answers', question: '" +encodeURIComponent(true_false_question)+ "' }",
success:...,
error:...
Note : the "{ }" parenthesis mean it's json.
Note2: url: "ajax_functions.aspx/webmethod1", add the web method to that page to handle the ajax call and data.
Note3: the web method signature must match exactly including case for 'data:' items.
e.g.
[WebMethod]
public void WebMethod1(string action, string question){
}