My guess is that you selected a template that uses endpoints as part of the line style.
Go to Window>Styles and click on the Edit tab. Click on the wireframe cube you see immediately below the Select tab. Untick Endpoints and also Extensions if that is ticked.
Do that in a new, blank drawing and then click on the thumbnail image at the top right of the Style dialog box. Immediately, without making any other changes go to File>Save as template and complete the save process. Make sure Use as Default is ticked when you do that and you won't see those "handles" any more.
I expect there are probably other things that are different with your new template as compared to the one you were using with the previous version. If you want, e-mail me a simple SKP file from your new version ( a rectangle drawn in will be enough and one from your old version, I'll figure out the difference and fix your new template.
