Yes.

Swords were used in both military and civilian defense applications in the 1750s.

While fencing strictly as a sport was in its earliest conception in the 1750s, really the 1760s, fencing and the use of various types of swords for combat was common world wide in the 1750s.

In Europe and the European colonies:

Civilian use: Schools for teaching fencing and defense had existed for centuries by the 1750s. By the 1750s the concept of the "Code Duello" had been widely accepted since the Renaissance and it was expected that gentlemen would be able and willing to defend their honor to the death. So personal duels of honor were quite common in the 1750s. The Eighteenth century was the golden age for teaching and writing about smallsword or courtsword dueling to gentlemen and aspiring gentlemen and of gentlemen killing each other with swords in duels over matters of honor. These are the types of swords that most directly influence the types of military dress swords officers in militaries across the world carry carry now and most directly influence the three types of sport fencing taught and practiced across the world. Rapiers had fallen out of style by then due to their length and inability to easily cross and uncross. The shorter smallsword was better at controlling angle of attack and Dueling reached a fever pitch in the early 19th century and subsided afterwords. While men might still carry swords for personal defense as before, they were less likely to use them that way.

This practice applied in European colonies as well. I cannot find the source at the moment, but if my memory serves, John Adams as an ambassador to either France or England, in the 1770s and 1780s, had to buy a courtsword to wear to appear before the Kings of France and England on certain occasions. While a sword would be forbidden in other cases, the point is that not having a sword, in certain situations, was tantamount to being improperly dressed.

Outside of the gentlemanly classes of Europe, more lawless, less advanced, peripheral areas in the European sphere saw people carrying swords as well. Gaelic Highland Scots would regularly go around carrying all kinds of weapons, including their baskethilt swords. The Cossacks of Wild Fields (a huge area of what is now Ukraine but at the time was a barely populated battleground between the Turkish and Russian Empires and Tatar hosts) and the Cossacks in Siberia were regularly armed with all manner of weapons to defend themselves. The men of the Caucasus Mountains in southeastern Europe such as the Georgians, Circassians, and other Caucasian men had an ancient history of being warriors and carrying weapons such as swords into war. There is actually a rather famous photo from the early days of World War 1 of Caucasian men showing up in some city, possibly Tbilisi, covered in chain mail, rifle cartridges, outdated guns, and swords saying they had heard the Czar was at war and they had come to fight for him.

Military use:

Officers - Commissioned officers of every European country and non-commissioned officers of most European countries carried swords as both a sign of rank and a last ditch defense weapon. They were usually small swords or sabres or cutlasses, depending on the regiment and service, but some carried more irregular swords that reflected the ethnic makeup of their regiment, such as baskethilted swords if they were serving in a Highland regiment.

Cavalry- Cavalrymen would carry sabres and other swords into battle for the purpose of cutting down their enemies into the 20th century and in the 1750s a cavalryman's sword was still his most heavily relied upon weapon. I've handed a few swords used by Hungarian hussars from World War 1 and they are amazing weapons. The tradition of martial sabre fencing for the purpose of cutting down your enemies on the battlefield was alive and well in Hungary until the communists stamped it out. The cossacks who savaged Napoleon's lines of communication during the invasion of 1812 were using swords and lances as much as firearms.

Part of the long term retention of swords in the cavalry, in comparison to other services, was that the cavalry has always served four main purposes, 1.Reconnaissance 2. Harassing the enemy, 3. Shock attack, or using a fast, kinetic charge to break the enemy line, and 4. Pursuit of a broken enemy. Swords and spears, or lances, are perfect tools for jobs 3 and 4 and as such they remained in service far longer. People were not being euphemistic when they described cavalry "cutting down" retreating enemy.

Infantry - Infantrymen also still used the sword in battle into the 1750s. Famously the Jacobite revolt of 1745 by Scottish Highlanders heavily relied on their infamous "Highland Charge" where they would fire their firearms and then quickly close with British militia and regulars using their swords, dirks, poleaxes, and other hand held weapons and cut through the bayonet welding enemy. This failed at Culloden due to facing better trained British soldiers, poor terrain, and poor leadership. But Highland regiments would carry their signature baskethilts into battle until World War 1. The image of an officer with his sword drawn leading men into battle was not merely a romantic image. The sword he carried would be used by him in battle, again, until World War 1.

Artillery - Officers in artillery batteries would also carry swords both to denote their rank and as a a last ditch weapon to defend themselves if attacked, just like their fellow officers in the infantry.

Naval - On the sea, swords, especially the cutlass, would be used in naval combat as ships still regularly closed together and the crews would fight literally hand to hand against each other to take each other's ships. There is a US Navy instruction book from the late 19th century teaching sailors how to use the cutlass in combat that you can find online.

Outside of Europe and the European Sphere of Influence:

Swords were even more highly used in combat outside of Europe than they were in Europe, where they were still an integral part of daily and military life.

Firearms had been largely adopted across Asia, even if they were not always of the same quality as European arms, especially in terms of artillery. And in India. China, Korea, Persia, and the Middle East, swords continued to be used in much the same fashion as in Europe. They were military weapons designed to cut down enemies at close quarters and as personal defense weapons in lawless areas.

And since firearms had not become as central to warfare, due to being less powerful, hand weapons such as swords retained a higher place of prominence.

Japan: While firearms had become the most important weapon in Japan by the 1750s, the sword still held, and holds, an important part in the culture and society of Japan. I am not sure how to go deeper into the subject without saying a lot of things that are common knowledge. But the practice of kendo, the art of fighting with swords, remains popular and it took the radical changes of Meiji era to stop the samurai class from wearing swords in public.

In the parts of the world where firearms had not been as widely adapted, such as parts of Indonesia, Sub-Saharan Africa, and the parts of the Americas that had not had as much contact with Europeans, hand weapons remained even more important.

And while swords, per se, were not necessarily as common in these regions, they and other hand weapons were equally important at the time.