Good question! The first example is according to certain news outlets, though it has never been adequately confirmed. Supposedly, in 2005, President George W. Bush proclaimed that God had sent him on a mission to fight the terrorists in Iraq and Afghanistan. However, President Bush is bound by the Constitution. The Constitution makes it clear that church and state shall be removed - in other words, that the reason for engaging in activity can not be religiously motivated. That's a bit of doublethink for you. We accept his decisions as our President, but we accept that he is disobeying Presidential law.
Another doublethink moment has been repeating. In the past 4 years, there has been a large push from doctors advocating that prison inmates be provided with needles for drug use. The doctors are suggesting that inmates are becoming more sick because they are re-using and/or sharing hypodermic needles. Therefore, the government should give clean needles to the inmates for their "safety." Give needles so that inmates can inject illegal and harmful drugs into their system... for their safety? Explain that one!
The Constitution does bound the reasons for engagement in the activity from religion, and in that sense President Bush is indeed a good example of doublethink-which we accept.
However, I do not really think the second example shows doublethink.