Sorry to hear that. Still I think that getting sober is the most important thing for you. If you put off rehab to be with your family, that would be understandable. But would you really be there for them, with all this stress about alcohol going on?
Maybe it's better to go to rehab first, get clean and then you can be there for them completely.
What I would do is to discuss this with them. Tell them why you are doing what you choose to do and that it's with their well-being in mind as well. In the long run, the sober you will be able to support your family on all sorts of matters much better. And in the good times, you'll be a better influence as well.
If at all possible, put your sobriety first. It's an investment that will pay of in the long run.
Sending you love and strength!