Exotoxins are plasmid-encoded. You can acquire plasmid genes via conjugation or other methods very easily. It's likely that some ETEC strains have LT and others ST based on their environment and interactions. That being said, can a particular strain have both? Probably. But if you were to eat contaminated lettuce in Mexico (i.e. traveler's diarrhea with LT), for instance, it would only require one bacterium to multiply at room temperature for half a day to generate sufficient LT to get you sick. So in terms of clinical relevance, it's likely that people tend to experience one or the other, but not both, at any one time.
Good question though. And no, I'm not sure as to whether a double-encoded ST-/LT-plasmid would cause one toxin to dominate the effects of the other.