Welcome.
Let me first assure you that you seem to have been unlucky. I also would consider whether you actually
like Cuban sandwiches. Many good people do not. They can be very dry by nature (though I have heard people complain about them being too greasy because butter is often used, so go figure) and should be had with a cafe con leche or a potage such as the very good garbanzos at La Unica or a fabada or caldo gallego, etc. I often get a medianoche if the soft, yellow egg bread is available. Same sandwich, different (less dry) bread.
Moros y Cristianos, sometimes also called congri, are likely the "black beans mixed with rice" you had, unless they simply poured black beans onto white rice. In point of fact, moros are not simply "black beans and rice mixed together"; they are cooked together with pork, olive oil, garlic and other ingredients to make a "sofrito" in a cheap but rather involved recipe. Cubans, including those at La Unica never, ever mix "regular" frijoles negros with rice before serving. I won't bore you with the recipe for moros, but I will vouch for La Unica's version. Again, correct moros are rather dry. One needs to be eating something else that is "wet" such as picadillo or ropa vieja. I'd say a Cuban and a plate of moros is a recipe for disaster or at least extreme dryness at a Cuban joint.
Regarding the pork, La Unica absolutely makes some of the best lechon in Chicago, selling it by the pound on weekends. It has fat, is fatty, not unlike Mexican carnitas which are slowly boiled in lard. That's how it is. They do not "gringo down" the food, though they might make one an extra-lean sandwich if asked. I'd say it is pretty odd to get only fat, and not much of it, to paraphrase Woody Allen.
Oh, and the French/Torta type bread is pan Cubano, Cuban bread. (And, a torta is a sandwich, at least in Mexico. The bread for a torta can be a bolillo, telera, pambazo, semita, etc.) Though La Unica's Cuban bread is not up to the standards of South Florida Cuban bread, it is a game approximation baked by a Mexican or Colombian bakery to La Unica's specifications.
Lest it seem like I'm being overly defensive or critical on La Unica's behalf, let me say I am trying not to be. In any event, La Unica, well known here and in the Latin community, hardly needs my defense. More importantly, based on my own experience, La Unica is "authentic" and very solid in the kitchen most days. Assuming that you do not simply dislike this type of food in general, your poor experience is unusual.
On a more positive note, please identify for us the other pork sandwiches you mentioned you have had and enjoyed around town.