Could you verify to see if the bold parts sound okay? I intentionally didn't backshift them. (I know backshifted tense would be most natural in a case like these, but I just want to know if these are possible)
A: You know, Gary doesn't seem to trust Betsy is sick.
B: Leave it to me. I'll talk to him.
(A moment later)
B: He believed/trusted she is sick.
A: Good. She is legitimately absent today.
A: Gee, John never believes me about anything I say! I'm telling him Betsy loves Alex, but he never seems to believe me! I'm telling him the truth.
B: I'll see what I can do for you, okay?
(B goes out, and comes back a minute later)
B: He believed/trusted me Betsy loves Alex.
A: Good thanks!
In both cases, not backshifting is fine. I don't like the use of trust though in either, though especially in the first. We say 'Trust me- she loves him', but I wouldn't use it as a reporting verb like this.