Both of them are fairly off-base, in fact.

Plenty could be said on the leadup to the assassination, but I'll focus on the day itself, first by giving an outline of what took place, and then addressing where the two accounts you gave in your OP were off.

The point of multiple assigns is, in fact, accurate. In addition to Princip, there were five other assassins along the parade route. The first two, however, failed to use either their pistols or bombs as the parade route passed.

The third assassin, Nedeljko Čabrinović, managed to throw his bomb as the motorcade passed. Rather than succeeding in killing the Archduke, however, it bounced off of the lowered convertible roof and exploded under the next car. A number were wounded and the car was knocked out, but no one was killed, the Archduke's vehicle remained intact, and Čabrinović failed to commit suicide both because the cyanide he had been given was very low quality and because the river he tried to jump into wasn't at all deep enough. He was captured shortly after.

Some of the other assassins, hearing the explosion, believed the operation was already a success and simply attempted to get out of there and hide their weapons. The Archduke reached the City Hall of Sarejevo, his destination, safely.

There the Archduke, initially irate and interrupting an introduction by the mayor, managed to calm down and give his speech, adding that he was grateful to see the joy shown by the locals at the failure of the attempt.

Following the ceremonies there, the initial plan was altered and the Archduke and Archduchess decided that they would go to the hospital to visit those who had been wounded during the bombing. The route was changed in order to avoid the city center and any further attempts, but these changes were not relayed to the drivers, which proved to be a fatal mistake.

Princip, meanwhile, had been contemplating finding a new position after learning of the failure of the first attempt, and did so near the Latin Bridge. This just so happened to be at a divergence between the original route and the newly changed hospital route, and as the uninformed driver took the wrong turn, he was forced to stop the car in order to turn around. It was here, stopped near Princip, that the fatal two shots were fired: one to the jugular vein of the Archduke, the other to the abdomen of the Archduchess. Both died en-route to treatment.

Now, as for the accounts you gave, these ones stand out as wrong.

Franz was shielded from the first assassin's bullet by his very overweight wife.

To begin with, there was no "first assassin's bullet", as before Princip the only actual attack was the throwing of a bomb, which bounced off the convertable cover. Furthermore, while I don't know about her weight itself, I've never heard Sophie referred to as 'very overweight', or at least any more so than her husband. A simple image search confirms that somewhat.

When the driver of the wagon realized there was an assassination attempt in progress he supposedly panicked and continued on the parade route path at an increased speed instead of veering off course. This allowed the subsequent assassins to take their shots. We were told the driver was held in high suspicion due to this.

Again, this didn't happen and it seems to me that your teacher was of the belief that the assassination occurred on a single drive, rather than having the initial drive to the City Hall, with the killing taking place later. This lines up with the final part from that account as well. I've also never heard of any claims that the Archduke's driver was under suspicion of being a conspirator.

From Dan Carlin:

After the initial attack the driver did not continue on the parade route but went to the local magistrates office on Franz's orders.

They did continue on the parade route actually, simply going faster than originally planned. The local magistrate's office, that is, the Town Hall, was the original intended destination.

The subsequent assassins on the route packed it up and (at least in Princip's case) went out for lunch. Franz and wife were shot and killed later in the day by Princip after he finished his sandwich.

The legend of the sandwich is a fairly common one, but a legend at that. It has, in fact, been asked several times on this sub, and probably stems from the fact that Princip was outside of a delicatessen near the Latin Bridge as he waited for the motorcade to return. Sandwiches themselves, aren't really a part of the local cuisine. A good article thats been linked on that subject several times in this sub can be found here.

I'm using the only source I've got on my desk at the moment for this, Sleepwalkers by Christopher Clark. Its a much broader, in-depth overview about the causes of the war in general, but has a pretty thorough account of the events of the 28th of June.

Professor in college told the following story about the assassination.

The Archduke was extremely vain and his uniform was tailored to his body. The bullet had hit his jugular but the severity of the wound was not clear until they cut the shirt out around his neck. The tightness had served somewhat as a tourniquet. When it was cut, the blood gushed and the Archduke died.

Any truth to that?

While the reality differs in some details from the account you cite above, I'm wondering if your teacher might have gotten the assassination of Franz Ferdinand mixed up with that of the Russian Tsar Alexander II.

In 1881, while indeed riding in a closed carriage through the streets of St. Petersburg, the Tsar was assailed by the first of three waiting assassins. This one threw a small bomb which exploded beneath the sturdily built carriage, killing many of those aboard but sparing the Tsar for the time being. The chief of police, who was travelling with the Tsar, tried to persuade him to flee at once, but Alexander demanded to stay to tend to the wounded and examine the scene in person.

A second assassin, likely scarcely able to believe his luck, threw yet another bomb. This one did its job, and the broken emperor fell in the streets, mortally wounded. He would die of his wounds later that day.

A third assassin hovered nearby, but his services did not prove necessary.