In my mind there are different types of cognative dissonance.
The two examples in the post are both examples of values v. logic. Or F v. T.
It is also possible to have values v. values dissonance. To merge your two examples: Say you believe strongly in gay rights, but you also believe strongly in your faith (as defined by your church) and your church says being gay is a sin. Which value do you go with?
Another type is logic v. logic. There are studies that say soy is great for our health and we should eat more of it. There are studies that say soy is bad and we should not eat it at all. Both arguments are presented logically and have scientific studies, which one do you choose?
The first type, in my mind is hardest because it forces you to choose between values and logic. Strong T/weak F would just abandon "illogical" values. Example: athiests who gave up their faith because it was not logical. Strong F/weak T would just ignore any logic or scientific evedence that contradicted their faith. This seems common in fundamentalist Christians. If your T and F are reasonably close, then you would work with the situation, adjusting values or logic or both till they fit. You might try out many different adjustments searching for something that is acceptable to you. It may be possible that you cannot make them both fit and have to give up one, but it is more of a last resort.
The second type can be a very angst ridden choice but it is more straighforward. Which of the two do you value the highest. You keep the one you value highest. The other one you either get rid or modify. In the above example, you might switch to a more liberal church, modifying your beliefs, but not abandoning them altogther.
The choice between two types of logic is not as emotional as a values choice. It can still be very distressing though, since unlike values, there is an expectation that truth is objective and therefor logical things should be compatible. Both agreements would be looked at logically and objectively to see which one is more likely to be true. There is more of a tendancy to completely reject one agrument than adjust them to accomidate both, although sometimes you can find something that works. For example you might conclude soy is good up to a certain amount, but bad in extreme high levels.