Doom would have a fear of being vulnerable and its not him being a coward, I think some people took that scene in Infamous Ironman where Cynthia calls Victor a coward for not killing the rabbit and just think he is one? I'm not sure, but like in the end I just feel the Doom series ending was very rushed. As for his face, it boils down to either:
1. Doom cannot heal himself because his scars were of a mystical nature, cursed by Mephisto perhaps or whatever that demon hood guy was the last time he got scars.
2. Doom refuses to heal himself wanting to keep his scars as a reminder of the past.
Also I have an issue with people assuming Doom would run to Strange for help at any given time, while they now have a kind of friendship I don't believe they are friends in anything more than two beings who respect each other's mystical power, and formed a bond over that. Even in T&T Doom planned things out so that Strange would have to help him, it doesn't cross his mind that Stephen would have helped if he asked. Victor has a hard time asking for help, yet another one of his foils, he either preferences to have people indebted to him or finds a way to break even where he is not indebted to the other. So even if Strange has the power to heal Victor's scars I still think Victor wouldn't go to him.