Sounds like your husband is a keeper. My opinion (unwarranted & worth what you are paying for it} is that you are just having a little mild depression and really shouldn't make any drastic decisions for a while. I'm sure that you are more than a little upset about your kids problems, (and who wouldn't be) but I don't think gettting rid of Kizzy would solve anything, and probably only make you feel worse.
Best wishes to you and your family. I know its trite but try to focus on the good things, your arm is healing, your husband loves you & your dogs, your kids are well and making an effort to keep things peaceful. I'm having a hard time not singing " Count your blessings, name them one by one, count your blessings see what God hath done, count your many blessings, name them one by one & it will surprise you what the Lord hath done".
One of those blessings for you & everyone here is that you really don't have to hear me sing. One of my blessings was not talent in the music dept.