These are press to fit. So extraction is special press. The extraction may damage the pusher, but might not.
I did pick up some NOS for my speedmaster but not from Omega; all models take the same pusher. But I did pick up the last 4 in Sydney Australia.
I would remove the movement. Place the case in a soapy warm water solution and work the pushers to remove or dislocate the DNA and other gunk that works its way under the pusher cap. Use an artist fine paint brush to aid in the cleaning. Then find some liquid or spray silicon at the automotive shop for spraying your rubber door trim and trunk seals.
I place a bunch of this (thimble full) in a jar. Then using the paintbrush and having dried the case thoroughly, apply a good drop under the pusher cap on the exterior and then another drop on the interior of the case at the pusher. Work the pushers and gently warm the case gently with a hair dryer. Repeat and work on the pushers. Finish off with silicon grease (Seiko) or I use from the electronics shop; dielectric clear silicon grease (big tube and cheaper). Apply under the cap with the brush and the interior also.
Keep working the pushers.
This has solved many Seiko, omega and rescently two Zenith Futur that I have got running.
Cheers, Geoff.