Should I send an email to the address of my former professor that wasn't explicitly given, but google out myself?
From the formulation of your question, to me it seems your main concern is if it is polite to send an e-mail to the address you were never explicitly given, but instead found it on your own Googling.

It looks like you found his/her new work address. If that's the case, I'd say it's completely fine to send an e-mail. All the right reasons for contacting your former adviser before the deadline are already mentioned in @aeismail's answer.

I'd just like to add that I've been told over and over again that a good letter of recommendation from your former adviser is expected, and that it's extra great if you can have letters of recommendation from more than one of your former advisers: those are the people who were supposed to be working closest with you and who continually assessed your progress over a period of time. If you can't get them to say something good for you, you must be very very special in some other way to get accepted to a PhD.

Back to the is it polite point: it is a work e-mail you found. That's the way he's expecting to be contacted concerning scientific things. It's public and accessible to everybody. That does not mean he'll reply to everything, but if you're one of his former students, he should be glad to reply if you split on good terms.

Also, I've contacted everybody who gave me a letter of recommendation after I got my PhD grant, and I sad "thank you". I went to visit my supervisor in his office for a few minutes before I moved away for my PhD: those people, and he especially, are who made me love science, and made me look for a PhD in the first place. He often said that advising students on how to go further with science/research was one of the best parts of his job. So, maybe I was just lucky, but in my experience, your former advisers will be glad to help you, glad to know their help mattered and won't mind you contacting them.