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.

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

If $\pi$ and $e$ disagree in infinitely many digits, then there are continuum many choices of the particular subset of those digits to swap, and so we get continuum many different numbers this way. Since there are only countably many algebraic numbers, it would follow that most of the time, yes, you do get transcendental numbers by doing this.

I'm unsure, however, whether one can say that all the resulting reals are transcendental. Perhaps we'll have to wait for some number theory experts to answer.

Lastly, if it happens (as seems unlikely) that all but finitely many digits of $\pi$ and $e$ are the same, then $\pi-e$ would be rational, and furthermore swapping the digits doesn't actually do anything except on those finitely many digits of difference, and so this won't affect transcendentality. In this case, there are only finitely many possible reals resulting, but they are all differing from the original reals by only finitely many digits, and so yes, they are all transcendental.