When you set up your two groups, did you check the Story Admin for each of them?? If so, at first glance I don't see a simple fix for this to turn the draft checkbox on for one group and not the other group. I would have to study the code's logic more to see if I can figure out how to do this. I'm working on something else so I don't have the time right now. In this case, you can force the draft box to be checked for everyone and then those you authorize to publish can uncheck the box. To do this you can add these two lines of code:
if ($A['group_id'] == 3)
$A['draft_flag'] = 1;
If you did NOT use Story Admin but checked the story.edit, story.submit, and/or story.moderate instead (which I don't think are working properly right now) for the non-publishing editor group, then you can force the check box on for them and off for everyone else. It's the exact same code but you'll change the group id number. To find this, go inside Group Manger and access the link of that group. The url will look like this:
http://www.yourdomainname.com/admin/group.php?mode=edit&grp_id=53
The grp_id is the number you're looking for here and will replace the 3 (which equates to Story Admin). In this particular case, the code could look like this:
if ($A['group_id'] == 53)
$A['draft_flag'] = 1;
Now, that you have your two lines, find this section under function storyeditor in the story.php file:
$result = DB_query ("SELECT group_id,perm_owner,perm_group,perm_members,perm_anon FROM {$_TABLES['topics']} WHERE tid = '{$A['tid']}'"

;
$T = DB_fetchArray ($result);
$A['group_id'] = $T['group_id'];
**** PLACE THE TWO LINES OF CODE HERE ***
$A['perm_owner'] = $T['perm_owner'];
$A['perm_group'] = $T['perm_group'];
Find this section under function storyeditor which is just a little further down on the same page :
$A['featured'] = 0;
$A['owner_id'] = $_USER['uid'];
$A['group_id'] = DB_getItem($_TABLES['groups'],'grp_id',"grp_name = 'Story Admin'"

;
**** PLACE THE TWO LINES OF CODE HERE ***
$A['perm_owner'] = 3;
$A['perm_group'] = 2;
That's it...the box will be checked. In first scenario above for everyone and in the second just for that one group.