So... 47 credits transfer to ME (worst case -- I bet you could get the Physics to transfer, or in the worst case you proficiency out of it), while 76 transfer to Applied Math. That's a difference of 29 credits, which is one year of school. If you can knock out the two Physics classes that's probably around 20-22 credits (Physics courses tend to be more than 3 credits each), which you could do in a semester and a summer. I think it's probably best to spend the extra year (at most) getting the degree you want.
MechE is going to do more for you than Applied Math. The courses you would be taking to finish off your Math degree are probably not going to be very practical. I'm just guessing, but I'd imagine you'd be taking a few proof courses as well as things like Diff Eq and Linear Algebra. Might not be what you're looking for.
From what I've seen, Applied Math is mainly useful if it's paired with something else -- some sort of application. My research advisor had her PhD in Applied Math which she used with her undergrad in Chemistry to go into Computational Chemistry research. Other people I've seen have double majors with Applied Math. You certainly could find job offers with the Math degree, but an Engineering degree is probably going to do more for you since it's kinda a "thing" right now.
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No amount of planning will ever replace dumb luck.