SSMS How do I open a new Query Window using Same Connection Instance of current Window
If I'm doing something wrong or missing some Point please let me know.


You're not missing anything.  SSMS always has a separate session per tab.


  it's SUCH a pain to run a script... and then from that same tab have
  to then write in other code when I don't want to edit/add any other
  code to the page that contains the script I ran.


In SSMS the Execute command (F5,Ctrl-e), always runs the highlighted text in the query window, if there is any.  

And so the normal way people use SSMS query windows here is to highlight sections of code to run, often separating the main script from the test/debug bits by using multi-line comments.

For instance when developing a stored procedure, then query window might look like: 

create or alter procedure SomeProc @param int
as
/*
  declare @p int = 12
  exec SomeProc @p
*/
begin
  ...
end


Then run the whole script to recreate the proc, and highlight the test harness lines and run to test it.