With no way to quantitatively distinguish the plausibility of several different explanations that haven't been falsified by experimentation, what do you do?

You criticize them to find flaws.

Do you use the equations that have always worked before, or wait until there's an explanation for why they work?

Of course, we do know that quantum phenomena are the result of previously fungible multiversal objects becoming fungible again. But assuming you didn't. Then you would use an explanation of what to do in the absence of a satisfactory explanation of the phenomena in question. We always act on such meta-theories.