How to Make My 30″ Cabinet Stable Without It Falling Apart?

I accidentally broke my bathroom mirror medicine cabinet and now I'm trying to rebuild it. This is only my third project ever, and I don't have any pre-made instructions to follow. My goal is to create a cabinet box and trim that match the original dimensions, while reusing the old doors.

So far, I made a cabinet box using 30-inch long 3/4-inch plywood strips that are 4 inches wide, glued together with wood glue and using simple 2 shoulder rabbet joints. I also created a 1/4-inch rabbet along the back and glued in a 1/4-inch plywood backing, clamping it overnight.

Now I'm working on the trim for the cabinet, using oak wood and mitered corners. The inside of the cabinet aligns with the inside of the mitered trim, but the trim hangs over a bit on the outside.

Here's my main worry: will gluing the face trim on and clamping it overnight be enough to ensure stability, or should I consider adding rabbet joints along the backing of the face trim for more support? Is there anything else I could be missing? I noticed that in the original cabinet, it looked like the corners might have been stapled, but I'm trying to avoid spending any extra money right now. Plus, those oak boards feel pretty heavy, and I'm starting to doubt whether my cabinet box can handle it. ? Any help would be greatly appreciated!