Snowden did not engage in civil disobedience in the vein of Gandhi/MLK/Thoreau. But that's fine. Civil disobedience is just one type of civil resistance. Civil disobedience only works against petty laws with limited punishment.
In this case, the legal consequences were dire. Can you name any act of civil disobedience (submitting oneself willingly to punishment as an act of disobedience) that carried the maximum possible punishment of death or life in prison? Gandhi, MLK and Thoreau broke laws where the punishment was just a few days/weeks/months. I am sure Snowden would gladly accept such a punishment if that was the choice, since the current alternative is to spend his whole life, in fear, in a foreign country.
What he did was different. He broke a law with dire repercussions in his attempt to expose constitutional violations. What you are suggesting is closer to telling the members of the White Rose Movement to willingly expose themselves to the law. Yes, I know: Godwin. A better US example would be demanding the FBI burglars to submit themselves to law. I happen to think the burglars did the right thing by not submitting themselves to the law then. Do you? How do you see Snowden as different from them?
Unlike the burglars, Snowden could not keep himself anonymous after the leak, since the NSA, unlike the case with the FBI burglary (since the burglars were completely unrelated to the FBI), would have quickly identified him. They sent someone to his house, almost immediately after the leak. So he had to run and go public. And if he ran to the only places that can resist US extradition, without going public as he did, he would have been easily labelled a spy. Going public made that charge not stick with most people. I believe that Snowden, once having chosen to expose the constitutional violations, had no real choices other than the ones he exercised.