In general, I don't see a big difference between the two options. In both you use the adapter object as a local variable (method scope) and you use "using... " statement. You also are not sharing nor passed the adapter object in the doProcessing() method. So it's very clear that you wont get problems there.
You also are not using transactions, and the DB calls are independent. I would evaluate what doProcessing do. If it's a long processing maybe it's better to use a "using..." statement before processing and another after processing.
As a programmer that doesn't have any clue what processing do, I would prefer this:
using(var adapter = new DataAccessAdapter)
{
var e = adapter.FetchEntity(..);
}
var e2 = generateXml(e);
using(var adapter = new DataAccessAdapter)
{
adapter.SaveEntity(e2);
}
doSomeProcessing();
using(var adapter = new DataAccessAdapter)
{
adapter.FetchEntityCollection(...);
}
That way, if I need to refactor in the future, it's more clear to me.