I really relate to this as I'm an empath too, though I only have myself to take care of and when I'm drained I don't have kids to worry about. I can take time to recharge, which makes it easier.
I've been thinking about this too. I think people are capable of changing, but only if they really want to. And I think for the most part empaths don't want to. Even when it gets difficult, I think it's a problem we want to deal with. It's a step up from being apathetic, self involved, oblivious to how others feel, and being free to hurt them. We want to be empaths because lots of the problems in the world are caused by people who lack empathy. We suffer, but in the end it is worth it, because it is also something that encourages (almost forces) us to help others.
Maybe it's not the same for everyone, but this is what I've experienced and seen in others. We cling to the label empath and love conecting with others. It's not a trait we dislike or see as a flaw. So that's why I don't think empaths can ever stop being empaths.
I do think it's necessary to learn how to have control over it. Setting boundaries or time aside to recharge, learning how to say no like you mentioned, and even finding ways to help people so the feeling of hopelessness lessens. Self care and allowing yourself to put your needs first sometimes is going to help immensely.
I've also been playing around with the idea that being an empath is a choice. I know people have different opinions on this, and I'm not sure where I stand yet. Some consider themselves lightworkers, as if they were sent into this world to help people. And I think that is beautiful. But I also think that it comes down to our values, and connection with the universe and all those in it. Maybe everyone is capable of picking up on other peoples vibrations like we do, but they aren't quite tuned into it. This would mean everyone's capable of being an empath. But still it's not a road that would be easy to come back from, which makes me think of the phrase, "Once you're awake, it's hard to do back to sleep."
I rambled on a bunch here, sorry if this isn't what you were looking for. Just got me thinking.
And to answer your questions, I don't think anyone should fight who they are, nor should they continue on a route that brings them suffering. Finding the balance is key.
$$$$$$
I'm glad it is making people think. In fact its @valorforfreedom who has made me stop and take a good look at myself. I might not agree and act on it all, but there is a lot that I will act on.
I'm not always so great at connecting, bit of a loner, as can't always handle what goes with it.