I've seen this question asked about 4e, and I think I've found the passage in the 5e PHB that answers the question, but I'd like to make sure I understand this correctly before I impose it on my game.
The question is can sheathing one weapon and drawing another be part of the same "free action", or does this take a full action?
The PHB (page 190)
You can also interact with one object or feature of the environment for free, during either your move or your action. For example […] you could draw your weapon as part of the same action you use to attack.
If you want to interact with a second object, you need to use your action. Some magic items and other special objects always require an action to use, as stated
in their descriptions.
As a bonus question, what do you think about a Rogue using their "use object" bonus action to draw a weapon after using their free action to sheath one, leaving a standard action and movement action left to be used.
Best Answer
Unfortunately, it would require your action. You can sheath a weapon or draw a weapon using your free object interaction, but not both. (Addendum: The Dual Wielder feat allows you to draw 2 weapons or sheath 2 weapons using a single object interaction. Note that this does nothing to fix your problem, it just seemed worth mentioning.)
A Rogue with the Thief archetype is absolutely free to use their Cunning Action to Use An Object on any object they choose (assuming they have the feature, obviously), and the scenario you describe is a perfectly legitimate use for Cunning Action.
However, there is another solution for all the non-Rogue players who want to switch weapons and attack in a single turn: Drop your weapon, don't sheathe it.
Dropping something requires no action at all, not even your free object interaction. So you can drop your weapon, use your free object interaction to draw a different weapon, and attack with that weapon on a single turn. And move too, if you want. Just remember to come pick up your dropped weapon later (probably at the end of the combat.)