I’ve used my credit card at stores like Target and Walgreens—what is the best way to ensure they remove my card number so it’s not exposed in a data breach?

You don't have any defense. Barring a legal notice and then pursuing that all your data is removed from their system (which itself would be questionable, since they would need to have your data on file for their audit and information archival purposes), there is nothing you can do about it.