I also, have difficulty with the conversation between Jason and the family.
I guess, it all comes down to what kind of environment the family was raised in. In the kind of world that you are painting, I can't see her being so credulous. Given the circumstances, I can see her being cowed enough to accept "because I said so" as an answer but to actually believe, right off the bat... ...it bugs me.
That kind of trust takes time to develop. Especially if it has to overcome the resistance of being in front of the boogie man from her childhood stories.
I can see passive acceptance on her part. Her family was just about to get murdered/raped/enslaved and the guy who saved them said that it was OK. She finds herself in a "less bad" situation now. Heck, even if he was human she would be wary of Jason. He's not the one who saved her children. In a world of slavers, monsters and ghouls, the thoughts running through her mind would revolve around "what do I need to do *right now* to keep my children safe". Everyone and everything is a threat to her children.
Gunny saved her and her children. He said that Jason wouldn't hurt them and he said to do what Jason said. That should be enough to get short term compliance that can evolve into trust.
I can understand you wanting to get back story in but I think that it was just too early for story time from Jason. I can see it happening later but not yet. Maybe, she warily does what she is told for now. After she has had a chance to see that Jason has no interest in eating her or her children, I can see her being curious.
She would be less afraid if Jason said something along the lines of "Look, if I hurt you, Gunny would kill me slow. Besides, I have plenty of food here and it's not like I'm going to be raping anyone." This assumes that Jason has to eat (he has to sleep, so safe bet) and that the immortal ghouls have no sexual function (else, there would be a lot of reproducing immortals in the world). That would address her two biggest fears.
Also, remember that while, "being a Marine" might be all you need to know to start forming an opinion about someone, most people who haven't hung out with Marines don't know what you know. Heck, I'm probably atypical of civilians because I've known enough people in the different services to have a pretty good idea about how they will react. Most people only know what they see on movies. Most current day people would react as sheeple in this situation and the Mother, in this case, had never even heard of a Marine.
If the pre-war era was a mythical paradise and she learned that Gunny was from pre-war, she might be absolutely trusting in an almost religious way. Given the proximity of the base, there might be tales about the "mythical" Marines coming to save everyone. In that case, she'd be treating him and everything he said as if Superman was saying it.