[Alastair E]

Sometimes, if a core is stuck and it looks like it'll break before shifting but the coil is otherwise good--try this--

Take another slug and screw this down/or upwards into the former--If a setting is found that then works--then great!

IF however--as always seems to be, the adjustment needs to go The Other way, remove the test-slug you tried and then use a Brass screw of same/similar thread, cut down to 'slug' size, with a slot cut into one end.
--I think Aluminium works as well for this--Could roll up some ally foil and try it...

The Brass slug will reduce the inductance as its screwed in up to the stuck slug and could well be enough to get that (whatever) inductance within tolerance for the circuit to operate normally--After all, Thats all we really need to do....
--All depends on your coil-former not being completely out of shape or too brittle, although some epoxy could help strengthen it up....

Just one of those mad things a very old engineer told me years ago, and Ive used it once on an old tube radio IF transformer....