Many congrats on passing! In most states, there are a number of ways to be sworn in. A common way is in a mass swearing-in ceremony. The ceremonies are usually held in hotels, convention centers, etc. Hordes of newly admitted lawyers hear some speeches, beam foolishly at one another, and then stand and repeat in unison, "I promise to faithfully uphold the laws of the Constitution and this great state, to the best of my abilities, even if my client forgets to pay me." Or something like that. :-)
I don't know where you live, but I highly doubt you're *required* to attend a mass swearing-in ceremony. Many of my friends across the nation were sworn in by judges, notaries public, and others with official standing. I was sworn in at the law office where I worked at the time; it was a quick and informal matter.
Concerning an applicaiton: you fill out a simple application so that the state bar association knows who you are, where you live, where you work, and how to contact you in order to harass you to pay your bar dues every year. :-)
I don't know, off-hand, of any states that require an attorney to vouch for a newly admitted lawyer. Perhaps you're thinking of the moral character fitness application? Those applications almost always require personal recommendations.
Since I don't know what state you're practicing in, I can't give more specific answers. The easiest way to sort this out is to call your state bar assocation and ask them.