Re. the older girl responding to commands meant for the new dog. This is partly why I learned the dog's name preceeds the command. The name catches the dog's attention and the dog is supposed to know to wait and see what the command will be. With more than one dog the name tells all the dogs who the command is meant for. If your girl did not learn this then you could either separate the dogs while training or, your girl might even be a help in training.
Separate training might be good for another reason too. If your new boy bonds more to your older girl than to you it's going to be that much harder on him to lose her when the cancer progresses.
I agree, give your girl lots and first attention and plenty of it on her own. Poor girl, is it possible the cancer or meds she is on for it make her feel out of sorts and dejected? She may have coped till the newcomer pushed her over the edge. Animals are very good at hiding their pain. Good wishes for her.