Is is possible for a Catholic to commit a mortal sin through ignorance?..
Yes, absolutely it's possible because sin, as an offense against God, is independent of our knowledge of the sin. For example, if a 3 yr old goes around using blasphemous language, his blasphemy is a mortal sin (because the act is a grave offense), even if he is unaware and not guilty. Blasphemy still offends God, regardless of our knowledge, because evil exists outside of ourselves.
If so, then are they fully culpable when they discover it along with the obligation to confess?..
In the case of the 3 yr old who blasphemes, of course they would not be required to confess such a thing, even when they made their first confession. In the case of an adult, who became aware of a mortal sin they did not know about, their guilt for the sin would depend on a variety of factors. Usually one should confess it at the next opportunity.
One has an obligation to have what's called a "properly formed" conscience, so that we know what is/isn't a mortal sin. If our conscience is ignorant in many areas, this is in itself a sin of laziness and omission. So, yes, it's very possible to commit a mortal sin "without full knowledge" if our conscience is ill-formed.