You want to indicate empathy and you are thinking in the right line. Funny thing about language is almost every word in the dictionary is either directly or in a specific context, subject to (mis)interpretation.
Would you really be sorry that your W would feel a certain way?
Would you be saying that you somehow feel responsible, so you are apologizing to her?
It is often suggested by members on this site to use the phrase:
+ "I understand how you feel that way about (insert topic)."
What you are saying is that you appreciate the reality that she may have a different perspective of something than you do. Nothing more, nothing less. It IS sincere because while she may say that your don't understand HOW she feels (which COULD be correct), that is NOT what you are saying because... well, because you DON'T feel what she is feeling. It might be the same as saying "I agree to disagree", but that statement is more aggressive and really indicates that you disagree so in some ways is saying that you think her opinion or feeling are somehow wrong...
You could stop right there. You do not have to dismiss yourself unless you feel yourself tensing up, suggesting that you may be prone to react soon. Get yourself away BEFORE you react. Have SOMETHING to do that you COULD do randomly, at any given time. That way, you won't be stuck with excusing yourself with something that could be a lie. Yes, it could be needing to go do laundry... or it could be going to the gym... don't use work as an excuse, because that is possibly one of the things that your W did not like, that you buried yourself in work to avoid her. Try to make it FUN stuff... or even things that your think might be interesting to her (but ONLY if it is interesting for you and something you want to do)...
IF you DO believe that it is some offence by you that might have been inappropriate, then by all means, apologize. If that is the case, then apologize ONCE and let it go. No one who is sincere in an apologize should need to apologize more than once. The apology should be sincere and then the offending individual should SHOW that they are sincere by not doing it again or "fixing" what ever behaviour was "broken".
Practice this stuff, NM... because you most likely WILL mess this up initially... but as time goes by and you practice this more and more (and you can use this with ANYONE, so opportunities to practice should be available to you), you will get better at it...
