If you're splitting territories using Path->Divide, just click on one of the newly formed territories and use the arrow keys to move it slightly to remove overlap.
Double borders while making maps: 2015-07-08 16:56:41
Yeah, I suggest simplifying the line (CTRL+L ?, I forget hot keys) first. reduce the number of nodes.
Then you can shift click on the line after doing the divide (selects the two nodes on either side of click) until you have the border you want to move selected. then just move that portion enough (usually thickness of line will do).
Double borders while making maps: 2015-07-08 17:03:38