I have replaced a lot of bungs in my life, especial in Teak
. My method is as follows.
First I take a cabinet scraper and bring the the surface to bare wood.
Then I take a machine punch about 75% of the diameter of the bung, and center it and give a sharp blow, this breaks any bond typically.
Then I use a drywall screw with the head snipped off chucked into a cordless drill. Use on slow low torque, and screw into the center of the bung. Now either the bung will back out or the screw will strip out the bung. Without damage to the underlying screw or the wood around the bung.
If the bung fails to remove it's self. Then my next operation is a forstner bit with the pilot ground off. I have an 3/8 thick SS plate machined to the fosrtner bit sizes which is 2" x 3" on the back side I have 80 grit sandpaper glued. Then I just center the guide on the bung and drill out the offending bung.
Going back make your new bungs with the tightest vertical gain wood of same species. Less swelling. Never drive the new bung to bottom end, as it will end up standing proud after you refinish do to swelling.
or shellac to set the new bungs.