For this the best solution is virtual cards. When you want to cancel, don’t argue with customer service about canceling. Just lock the card.
The only time virtual cards are particularly useful is if you're giving your CC# to some very shady merchant, probably one without any recourse if you decline payment. But, perhaps, just don't give those people your cc# in the first place.
If it isn't a virtual card your bank will probably cancel your existing card and issue you a new card with a new number. Some merchants where you had the old card on file might get the new number automatically via a credit card update service, but it is likely some will not and then you need to remember to go manually update them if you want them to keep working.
If you had used a virtual card on the site that got breached it should in theory not affect the underlying card or any other virtual cards derived from it.