So now I need to remove the older, smaller handyboxes and while I
managed the wiring part without damaging anything, I don't know how to
remove the existing handy-boxes to be able to install remodel boxes
without creating a mess. Is there a method that will allow the least
breakage of the drywall so that I can patch it up easily?

Do you reacll or HAVE another handybox similar to what you installed ? The
metal ones have a flange that can be adjusted backward intil the box can be
pulled out w/o any damage. Some of the blue plastic ones have two corner
tabs that flip up/out within the wall to secure the box. If you can figure
out where the "tabs" are, you can cut thru the drywall with something like a
sharp screwdriver so the tabs can be eased thru and out of the wall. You can
use some "mud" and f'glass drywall tape to fix the cuts once you get the box
out and then slide the new (deep) ones in.