Absolutely keep up with the neosporin, if nothing else your preventing infection, expediting the healing process, and keeping the scales from getting too dry and peeling off prematurely exposing the area too soon.
In terms of the thermostat temp, your "hot side" temps should be around 90-93 degrees for the UTH. 83 is a bit low, that's the correct ambient temp for the snake but you should give your snake the option for a hotter area to hang out if he/she desires it.
From what it sounds like to me, this "dry blood" is just the scales discoloring from them dying off making way for new scales to come. Unless you see actual wet blood i wouldn't worry about that. I would just avoid contact with that area as much as you can, but again keep up with the neosporin that can do nothing but help at this point.
Hope everything is ok!! Keep me posted