There are two possibilities that we can consider here. The first, as you brought up, is the False Dichotomy Fallacy. Our person assumes there is only two possibilities, A and B, so after eliminating A she believes she has proven B.
The second fallacy is referred to as the Blind Loyalty Fallacy. Our person assumes their source cannot be misinformed/mistaken, so after eliminating the possiblility that he is lying, she concludes that what he says must be true.
We can agree that we probably started with a trichotomy (or some other finite number of possibilities).
- Our source is lying
- Our source is mistaken
- Our source is correct
The first logical misstep that the person makes is assuming that our source cannot be mistaken and thus eliminating option 2, a clear example of Blind Loyalty Fallacy. This fallacy then causes what we could call a false dichotomy down the line, but the argument was clearly already ruined before this point.
Fallacies are incorrect lines of reasoning. At the point in the argument where we have eliminated the possibility that our source may be mistaken, it is in fact "logically sound" to assume the he must be correct since he isn't lying. Because of this, it would really be incorrect to call this a false dichotomy fallacy. The only real mistake in this person's argument was when he eliminated option 2 above, and thus it this is an example of a Blind Loyalty fallacy.