What is the current (2024) rule on swapping the hand you're holding a weapon with? 

Like if I'm holding a 1-handed melee weapon with my RIGHT hand and decide to start holding it with my LEFT hand. What do the rules say about that? 

RAW, does that count as an object interaction? Does it count as drawing the weapon? Can I just do it without any cost at all? 

Please cite to the text of the new rules or an old rule that is confirmed to be unchanged. Thank you. 

Transferring objects between hands isn't specifically defined in the rules. A DM can determine it may be done freely, count as equipping/unequipping a weapon or as an object interaction.

The game's mechanics don't get down to the level of granularity of handedness. You're holding a weapon in a hand, but nothing forces you to specify which one.

If your DM has decided to care about it for some reason, you're going to have to ask them for the ruling.

If you hold an object in one hand, nothing stop you from picking up an object in the other hand, provided it's free.

If for some reasons you absolutely need to use the right hand for something and it isn't free, the pertinent rules are:

Interacting with Things. You can interact with one object or feature of the environment for free, during either your move or action. For example, you could open a door during your move as you stride toward a foe. If you want to interact with a second object, you need to take the Utilize action. Some magic items and other special objects always require an action to use, as stated in their descriptions.

Equipping and Unequipping Weapons. You can either equip or unequip one weapon when you make an attack as part of this action. You do so either before or after the attack. If you equip a weapon before an attack, you don’t need to use it for that attack. Equipping a weapon includes drawing it from a sheath or picking it up. Unequipping a weapon includes sheathing, stowing, or dropping it.