In the Player's Handbook (PHB), the rules for taking the Dash action read:

When you take the Dash action, you gain extra movement for the current
turn. The increase equals your speed, after applying any modifiers. —PHB, p. 192

The rules also provide a list of "the sorts of things you can do in tandem with your movement and action" (PHB, p. 190; Interacting with Objects Around You sidebar), including the ability to "draw or sheathe a sword".

Can a PC draw/sheathe a weapon during his/her movement AND draw/sheathe a weapon during a Dash action taken on the same turn?

For example:

Example 1

Turn 1

Action

Ranger fires longbow at enemy from a long distance.

Movement

Ranger stows longbow and begins to approach enemy.

Turn 2

Movement

Ranger draws scimitar with right hand while continuing to approach enemy.

Action

Ranger uses the Dash action while drawing a second scimitar.

Example 2

Turn 1

Movement

Rogue approaches enemy from a distance.

Action

Rogue fires crossbow.

Bonus Action

Rogue Dashes and stows crossbow. (A Rogue's Cunning Action feature [PHB, p. 96] enables them to Dash, Disengage, or Hide as a bonus action.)

Turn 2

Movement

Rogue draws shortsword while approaching enemy.

Bonus Action

Rogue Dashes up to enemy while drawing a second shortsword.

Action

Rogue attacks with shortswords.

You can interact with one object for free; any more requires the Use an Object action

The "Other Activity on Your Turn" section describes the ability to interact with one object or environmental feature for free:

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 open a door during your move as you stride toward a foe, or 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.

The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle. For instance, the DM could reasonably expect you to use an action to open a stuck door or turn a crank to lower a drawbridge.

Drawing or sheathing a weapon is one of the things you can do as an object interaction. However, you can't take multiple object interactions for free on the same turn. In order to do that, as the rule states, you would need to take the Use an Object action:

You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn.

Even if you both move and take an action (any action), you still only have only one free object interaction per turn. Any more than that requires the Use an Object action. And unless you are a Thief rogue, you must use your action to do it (Thief rogues have the Fast Hands feature, which lets them use their bonus action to take the Use an Object action).

No, you can't.

As the other answers have said, this doesn't generally work without using an action.

However, since both of your examples include trying to draw two weapons in the same turn, it might be worth noting that the Dual Wielder feat (PHB, p. 165) confers the following benefit (among others):

You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.

This does not work.

You get one free interaction with objects on the turn that you can use during your move or action, as described on page 190 of the PHB:

Other activity on your turn

[...] 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 open a door during your move as you stride toward a foe, or 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.

You can draw a weapon or sheathe/stow a weapon during either your movement or action, not both.