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The question has been edited since this answer was first posted. Originally, the title question asked specifically about map cases. There will be further updates.
Yes, they did.
One such bag was the Sabretache,
a flat, leather pouch or satchel with long straps traditionally worn by some cavalry and horse artillery officers from the left-hand side of the waist belt near to the officer’s sabre...
Sabretaches ultimately derive from Hungarian haversacks or bags (called tarsoly), which over time became more elaborate. Such bags were used for keeping fire-making tools and other, small essential items. In the eighteenth century, Hungarian Hussar uniforms and accessories (including sabretaches) became the ‘standard’ form worn by certain regiments of cavalry officers, including those of the British, French, Polish and Russian armies.
The Sabretache of Jean-Jacques Desvaux de Saint-Maurice (1775-1815), French general. Attribution: Aclaudon [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]
"Hussar of the Magdeburg Hussar Commando (1763)" Source: Wikipedia
Officers might also have had simple knapsacks such as the one used by Captain Uhl, a militia officer during the American Revolution.
"Knapsack of Captain David Uhl, Dutchess County Militia" Source: scribd, via pinterest
The earliest map case I've been able to find is this French one from the mid 18th century on the Military and Historical Image Bank.
Source: Military and Historical Image Bank
Unfortunately, there is no further information on this item.
Map cases were also used by Pony Express riders; a replica of a brass map case can seen here. It would seem that in both examples the map would be folded before being put in the case.
Maps were at least sometimes folded as the painting below by Édouard Detaille (1848 to 1912) showing Napoleon at the Siege of Toulon (1793) illustrates. Although Detaille was not at Toulon, he was
noted for his precision and realistic detail. He was regarded as the "semi-official artist of the French army"
Source: Napoleon à Toulon par Edouard Detaille
This is all somewhat tenuous of course, and there is no evidence of a map case in the painting; Napoleon may well have kept maps in his pocket.
Portmanteau
This would not, of course, be carried around by the officer on his person but it could be within easy reach. In addition to clothes, linen etc., officers could keep papers, maps, instruments in a portmanteau on a horse. The article Portmanteau states that
By the end of the eighteenth century a portmanteau is recommended as standard equipment for an army officer.
The officer would have three horses:
An officer would only have one servant and all his necessities would have to be carried on 3 horses. 2 portmanteaus were recommended. One carried by the servant's horse should contain all the necessary clothes and associated paraphernalia, the other, transported by the pack horse, could contain less essential items such as books, maps, papers etc.
The cavalry horse and his pack, embracing the practical details of cavalry service goes into considerable detail on what an officer might have within easy reach, if not actually on his person.