CAVEAT: I've never used InDesign, I'm not an Illustrator expert, and I've never
made anything for print.

However, looking at your postcard, I don't see anything that a vector drawing
program (such as Illustrator) can't do. You used the vector aspects of
Fireworks, to make it, so you should be able to do that in Illustrator. The CC
logo should be easy enough: convert to paths then join.

I do, however, remember reading something about opacity/transparency effects
not giving the expected results when printing, but I don't remember where I
read that. (As a Web person, it didn't really apply to me.) Anyway, my point
would be to check the results of your drop shadow carefully.