It really depends on context, and there is no fixed rule for deciding when to use what. Personally, I think of "adjust" as implying a smaller change than "adapt", which generally implies a realignment of goals. Parents of two children who are now expecting their third child would need to adjust to the demands of a new baby. Even a newlywed couple could adjust to parenthood, if they had been appropriately prepared for it. On the other hand, a previously sworn lifelong bachelor, who got married by chance and is now expecting to be a father, must adapt to fatherhood, because his goals and aspirations were aligned in a completely different direction before.
So depending on what kind of situation you're trying to convey, you'll have to choose the appropriate word based on context. Even so, it's unlikely that your intended nuance of meaning will be clear simply on the basis of your word choice -- you'll probably have to explain the context alongside.