If the new client needs you to start immediately, I would ask you existing clients: How urgent is this? I've had some things come up and I'm trying to prioritize. Will it impact you negatively to push back our deadline to ____?"
You may be pleasantly surprised to learn they really don't care. I have nearly killed myself over deadlines only to learn later they were pretty arbitrary on the client's part, and the client was clueless that I stayed up all night. When I hire freelancers I let them know if something is not urgent because I know what it's like to be the freelancer, freaking over deadlines.
Most clients aren't thinking about that but they must be aware you have other clients/projects and if they like you and your work so far they'll probably be perfectly happy to let you coast a week or two.
ALSO - one way to buy yourself time is to send them some questions that need to be answered before you can resume their work (and will take a little thought on their part). 9 times out of 10 they won't get back to you for some time unless the work is truly urgent. Maybe I'm a bad person but I have had great success with that technique - just bounce that ball back into their court and and get cranking on the new client's project.
FINALLY - do you have any personal days you can take off from your 9 to 6 job? If so, you could use that time to get things under control. May be a downer to use your time that way but it could also launch you into new work that may ultimately free you up to quit your day job...