Are the cabinets with or without a face frame.
Is the dwr. unit a single dwr or multiple dwrs?

Pictures would definitely help our understanding of your situation here.

I'll try to get photos taken but meanwhile the drawer unit is a box with a face frame. Yes, I could screw the face frames of the drawer unit and the cabinets together to support the front. The back is more of a problem.

If the dwr. is a single unit, it should be the depth of the rest of the base cabinets.
So, depending upon the width you should be able to attach to a stud.
If it is to narrow, then insert shims on each side, up and down, in the rear corner and using an angle drill, insert a screw on either side.

The unit is too narrow to be able to screw it from inside like a typical cabinet. But you gave me an idea. I can mount a cleat on the wall so that the top of the drawer unit is at the correct height. Then connect the front face frames together. That should work.

You screw into the sides of the drawer cabinet box from the adjacent cabinets, making sure your screws aren't too long. Unless I'm missing something here in your description??

It sounds like you may be talking about a desk drawer type thing with an opening below, not an entire base cabinet with drawers. is that correct?

Sounds like the cabinet sides on either side of the nook have to be clear because they're visible, so the issue is how to attach the drawers on those sides from the inside, right?

A diagram or pictures would really help us to help you.

While we're waiting for the OP's photos, here's one of a drawer install in process, in case it provides any ideas--if this is, in fact, what you're talking about.