Nice question, Erin. Here is one quick easy thing to say.

Since there are continuum many choices of the particular set of digits to swap, we get continuum many different numbers this way, and so since there are only countably many algebraic numbers, it follows that most of the time, yes, you do get transcendental numbers by doing this. 

But I'm not sure whether one can say that they are always transcendental. Perhaps we'll have to wait for some number theory experts to answer.