The theory regarding the introduction of high fructose corn syrup ("HFCS-55") is completely false, for two reasons.

REASON #1

The formula had already been changed before New Coke was even launched. We know this in a quite substantial and public way.

Bottlers in the United States had since 1921 been working under a contract that provided a.) the syrup used in bottling contain a minimum of 5.32 pounds of sugar per gallon b.) the maximum charge would be $1.30 per gallon and c.) the price could change based on market price of sugar (in other words, it adjusted for inflation).

Note that the price formula only adjust on sugar, so it didn't account for all inflation, and by the late 1970s Coca-Cola was starting to feel a squeeze, and proposed a "1978 Amendment" as a remedy to the contracts. 90% of bottlers signed as there was an extra amendment for if the "sugar element" as a generic item was subject to a change in market price; HFCS was definitely cheaper and already being used for other affiliated beverages (like Sprite) so it was clear Coca-Cola was thinking of this -- and two years later, in January of 1980, the bottlers who had signed the amendment started reaping monetary benefits as Coca-Cola switched to a 50/50 formula between sugar and HFCS.

1982 incidentally saw the introduction of Diet Coke, and a giant legal kerfuffle, as it was Coke's position that the newness of the beverage indicated any syrup could be made under a new contract, and the bottlers thought the old one still applied, and a protracted decade-long legal battle that's past the scope of this answer resulted. However, it is useful in that part of the reason we have such good information about the timing of HFCS is because of the business disclosures required as a part of the lawsuit.

Returning to the HFCS timing issue: the amount was increased to 75% in February 1984, and finally to 100% in November of that year. New Coke's launch was April 1985. Hence, the HFCS change was already complete a full six months before the New Coke circus.

REASON #2 (note: some swearing)

Despite what some accounts indicate, the backtracking on New Coke was only partial at first.

The executives of Coke were all in. In the early 80s Pepsi was destroying Coke in taste tests and Coke had been flagging in sales for a while. The Mexican executive Sergio Zyman (behind the 1982 success of Diet Coke) was tasked with finding a new formula for regular Coke, and this led to the development of 7X-100, which according to Coke's own taste tests performed much better than both traditional Coke and Pepsi (with new Coke vs. old Coke blind it was 55-45, when consumers were told the identification the gap actually increased to 61-39). They assumed this was the energy needed to take over the market, and were very concerned about "splitting the market" (essentially guaranteeing Pepsi taking the #1 sales spot) and wanted to do an complete swap.

I won't belabor the disaster of the April 1985 launch, but Coca-Cola received 5000 angry calls a day and Fidel Castro called it evidence of the failure of American capitalism.

My littele sisther is cring because coke changed and she sayed that shed is not going to stop cring every day unitl you cang back. . . . I am geting tryer of hearing her now if you don't chang I'll sue evne if I'm just 11.

An Air Force officer explained that he was originally planning to have his ashes buried in a Coca-Cola can but was now reconsidering, and a Coke delivery man in Georgia was attacked by an angry woman with an umbrella.

You bastard, you ruined it, it tastes like shit!

A nearby Pepsi driver laughed, and was then also berated:

You stay out of it! This is family business. Yours is worse than shit.

July 10 led to an announcement of the return of Classic Coke. This was important enough for a special announcement to be made on ABC interrupting the soap opera currently in progress.

All internal evidence -- including a book that a journalist had been tasked to write in the anticipation of the glorious launch of New Coke, and the nature of the book turned into something else and executives started becoming shy about interviews -- shows this really was an utterly unintended consequence. Sales did eventually boost up for Classic Coke in the long term, but the executives still believed in New Coke and believed it would eventually overtake Classic to win.

Both beverages were advertised at once. Classic Coke went for classic Americana ("Red, White & You") while New Coke tried to get consumers to "Catch the Wave" of the future, and went for (as the executive Brian Dyson said) "high-tech promotion with highly contemporary imagery". Max Headroom was enlisted as part of the pitch (you can watch one of these ads here).

Millions were wasted on the New Coke campaign, far more than Classic, but New Coke kept flagging, down to a 3% market share, when a final dagger was stabbed as McDonald's swapped formulas from New Coke to Classic.

SUMMARY

So, briefly:

1.) High-fructose corn syrup was being phased in already starting in 1980, and was fully phased in a full six months before New Coke.

2.) New Coke was not dropped when Classic Coke came in; the executives still believed New Coke would win and put money and energy into an ad campaign, but the public itself caused New Coke to die.

It is true the eventual turnaround led to a stronger Coke brand overall -- and despite $4 million being blown on the New Coke debacle, it was more than made up for it by the increased numbers around Classic Coke -- but it was essentially an unpredicted accident, and HFCS was used before the event with a fairly wide margin.

...

Sources:

Pendergrast, M. (2013). For God, Country, and Coca-Cola. Basic Books.

One of the many legal opinions in the Coca-Cola Bottling Co. v. Coca-Cola Co. case for month-by-month detail on the introduction of HFCS-55.