how do I remove dried caulking around millwork/wall

Hi, I need to repaint an interior wall (drywall) - I had to remove the millwork and the dried caulking is stuck to the wall...the new millwork is narrower so I need to get the dried/stuck caulking off without damaging the wall....any good suggestions

 

I cut the caulking with a sharp knife and they remove the molding. Then I try a painters tool or putty knife at a very shallow almost flat angle to scrape the remaining caulk off the wall. If the caulk was applied over a couple layers of paint sometimes one of the paint layers lets go and allows the caulk to come off without damaging the wall. If the caulk was put directly on the wall board or on a primer coat I often end up ripping off some of the paper requiring patching and repainting. I've tried removing the blade from a box cutter and lay it almost flat on the wall and shave the caulk off flush. It sorta works but the smooth texture of the shaved caulk does not match the surrounding so it needs a rolled coat of paint to get the subtle texture to match.

 

I wouldn't worry too much about removing it neatly, unless you do a perfect job - it will show. I'd just scrape it off, apply a thin coat of joint compound, sand [texture if needed] prime and paint