They are both unnatural actually, but ‘on’ is less awkward than ‘about’. I don’t trust myself. (Best, ‘that’ being clear from context) I don’t trust myself to … I don’t trust myself to be reliably on time with a start time of 4am. I don’t trust myself to not hold a grudge. I don’t trust myself to do that. I don’t trust myself to remember that. I don’t trust myself to behave like that

Both are correct. The difference is the degree of precision. "On" is precise. If the cat is "on the table", you know exactly where the cat is. "About" is approximate. If the cat is "about the table", you know the cat is somewhere near the table.

I would say that "I don't trust myself on that." sounds more natural.