Also, don’t say, “I wasn’t aware it was so important to you.” That’s definitely a trigger phrase.

Try, “Thank you for letting me know this is important for you,” (or “how you feel”) instead.

A lot of the response here has been around the way the 'apology' focusses too much on the person who's supposed to be receiving the apology and not the person who has something to apologise for. The intended draft follows along the classic lines of "sorry if you feel that way" which implies that the person being apologized to is really the one in the wrong for having taken offence and the apologizer is just indignant at being forced to say anything rather than actually sorry for anything.

I get all that, but... Is there no way to sincerely express being sorry for not considering or anticipating another person's individual response to something as opposed to the thing itself? Without seemingly blaming them for that response? It's still about the apologiser's actions in having been inconsiderate in their deployment of language then, just not for the actual language. I ask because your proposed change "I'm going to work on improving my language" implies that the error was in using the word fuck at all and that their language is in general faulty in some way. I don't think that's the case. Having a manner of speech that includes that word is not something inherently bad, the bad behaviour necessitating an apology as I see it is for being too presumptive in assuming this particular person would have no problem with it when it's known that some people might and also for not immediately taking that person's offence seriously in the immediate aftermath when they expressed having taken offence (they didn't take it seriously, this is a follow up post).

It seems reasonable, if expressed very carefully, to commit to avoiding the word around them, since that's all that person can reasonably want, that's the problematic behaviour that is getting in the way of their working together. Committing to improving their language can really only mean committing to not saying that word generally which is defacto suggesting the word itself, not the lack of consideration is the problem and also puts OP in a position now of being on the hook in future not just for using the word around this individual but in all other circumstances as well something they shouldn't promise. If the work environment is such that nobody else speaks like this and they're the only one then sure, it should have been common sense to begin with and such a commitment is a no brainer, but if it's otherwise common practice and it's just this one person they need to accommodate then that's what should be done, accommodating this one specific person in order not to offend in future and apologising in order to let them know that you hear them and consider their feelings important.