The first actual test of the new election system didn't go so great. 1789 and 1792 don't count, since you don't get credit for everything going swimmingly when there's only one candidate. After Washington retired, though, his vice president John Adams wanted to take his place, which set up a tense race, since he wasn't the only one. The major opposing candidate was Thomas Jefferson, who used to be secretary of state, but quit after Washington kept favoring the Federalists Adams and Treasury Secretary Alexander Hamilton instead of him.
When Washington ran, the general agreement was that he would be president and John Adams would be vice president, and since everyone was on board with that, having every elector vote for him and most (but not all) of the electors vote for John Adams wasn't too difficult to arrange. The Adams-Jefferson race changed that, since barely more than half the electors favored Adams, which didn't leave much room for error.
The fundamental problem was that the electors couldn't vote separately for the president and vice president. As long as someone got a majority of the votes, that person was elected president, and the runner-up was vice president. If there were a tie, it would go to the House, and while this wouldn't be quite as embarrassing the third time around, it still wouldn't be good, and could get messed up pretty easily. So both Federalists and Republicans tried to arrange for all their electors to vote for the presidential candidate (Adams or Jefferson), and a couple to vote for someone who wasn't the vice presidential candidate (Thomas Pinckney or Aaron Burr).
This scheme failed pretty massively for both sides; Adams got 71 votes, but Pinckney managed 59, and he only managed that much because South Carolina, ordinarily a Republican state, cast 8 votes for him and Jefferson. Jefferson, meanwhile, managed 68 votes, and Burr was driven down to 30. Now, this seems pretty embarrassing, but remember that this was 1796, and they didn't have Skype, email, phones, or even the Pony Express. Good organization requires communication, and you try coordinating the actions of seventy people spread out over a thousand-mile coastline over the course of a month (the College had to meet on December 15, and the election was in early November) when your only option for telling them whom to vote for is a horse.
So Adams was elected president, and Jefferson accompanied him to DC as the vice president, followed by four solid years of bickering. The existing system for elections clearly didn't work too well, but Congress wasn't too interested in changing it. Rep. William Smith of South Carolina proposed an amendment distinguishing between ballots for president and ballots for vice president, but nothing happened. Would 1800 be any less chaotic than 1796?