Tamara, I am not sure what you mean by "our house is in both our names UNTIL TODAY." What happened today? Assuming nothing happened and it is a figure of speech, federal law allows heirs to continue paying the mortgage without the risk of acceleration. There is no reason to put your name on the loan and, besides, you don't want to add to your liability. As far as title is concerned, you should consult with a local attorney to review the facts and make recommendations as to using the best way accomplish your goal.
The title really doesn't matter--and there is no reason for you to jump on the mortgage hook.
You can file an affidavit of death of joint tenant at any time. As long as the payments are made, the house won't be taken from you.