My employer required that I have my new social security card before they would process any paperwork, so you may not be able do that in advance, as the article advises.
Also "With these two important documents out of the way, you should have no trouble changing your name on your bank accounts, credit cards, passport, and insurance policies" is a bit misleading. You won't have "trouble," but is is a very
time-consuming process.
You must visit your bank IN PERSON with your new drivers license and a certified copy of your marriage license in order to change your name on your bank accounts, order new checks and get a new debit card and/or bank-issued credit cards. They
will not do this except in person.
Many places will not do this on the phone or internet, it has to be done by mail or in person, because they need a new signature on file.
Replacing your passport requires new passport photos and surrendering your current passport, so make sure there's enough time before you need to use it.
Other key things to remember are the title on your car if you own it, your car registration, and all other bills/utilities that are in your name.
Good luck!