I have two questions about my Oathbreaker paladin's relationship to the undead.

When I use my Control Undead Channel Divinity option, I can take control of an undead. My question is "Does the turn undead resistance offer advantage on the saving throw?"

I tried to control a ghast one time. It had turn undead resistance so my DM said it had advantage on the saving throw because of that.

Is that how it works?

Directly related to the above is: when I use control undead again on an undead that I already control, does it have to make the saving throw again, or do I have an advantage somehow because I already control him?

The ghast would not get advantage against Control Undead

The ghast has the Turning Defiance feature:

Turning Defiance. The ghast and any ghouls within 30 feet of it have advantage on saving throws against effects that turn undead.

However, Control Undead, although it is a Channel Divinity power like Turn Undead, does not turn undead. Therefore, the ghast does not get advantage against Control Undead RAW. Features that turn undead will say so in their description, such as a cleric's Turn Undead or Devotion paladin's Turn the Unholy features, both of which say "If the creature fails its saving throw, it is turned for 1 minute or until it takes damage."

No advantage for attempting to control an already controlled undead

The Oathbreaker archetype (DMG, p. 97) including this on the Control Undead feature:

Control Undead. As an action, the paladin targets one undead creature he or she can see within 30 feet of him or her. The target must make a Wisdom saving throw. One a failed save, the target must obey the paladin's commands for the next 24 hours, or until the paladin uses this Channel Divinity option again. An undead whose challenge rating is equal to or greater than the paladin's level is immune to this effect.

Although it says that the undead must obey the paladin's commands, a saving throw isn't really a command, so you wouldn't necessarily be able to "order" the undead to fail it (see Can you choose to fail a saving throw?; in short, Jeremy Crawford tweeted "No rule lets you opt to fail a save. As DM, I might allow it, assuming you aren't incapacitated or dominated.")

Furthermore, to use Control Undead again on the same undead ends the control you already have over the undead. What that means is you have control (assuming we're still within the 24 hours since the first time you used it) right up until you use Control Undead again, then as you use it, in that instant, you do not control the undead, so it would not be affected by the previous use of Control Undead.

Control Undead is not a "Turn" effect

Strictly speaking, the Control Undead feature isn't the same as "Turn Undead". Other features that turn undead (or other creature types) are pretty explicit about what they do. Take the Devotion Paladin's Channel Divinity, Turn the Unholy, as an example:

As an action, you present your holy symbol and speak a prayer censuring fiends and undead, using your Channel Divinity. Each fiend or undead that can see or hear you within 30 feet of you must make a Wisdom saving throw. If the creature fails its saving throw, it is turned for 1 minute or until it takes damage.

A turned creature must spend its turns...

Emphasis mine. The feature as written for Clerics is also identically written.

By contrast, the Control Undead feature makes no reference to "Turning" the Undead it affects.

As an action, the paladin targets one undead creature he or she can see within 30 feet of him or her. The target must make a Wisdom saving throw. On a failed save, the target must obey the paladin's commands for the next 24 hours, or until the paladin uses this Channel Divinity option again. An undead whose challenge rating is equal to or greater than the paladin's level is immune to this effect.

Therefore, I don't think it's appropriate, in a strictly RAW ruling, to apply the Turn Resistance or Turn Defiance that creatures like the Topi or Ghast have to this ability.

Control Undead is very similar to Turn Effects

Although I don't think it's appropriate to treat these effects identically RAW, I do understand why your DM grouped them together. Virtually all other Turn effects that are in this game come from Channel Divinity Options provided to Clerics and Paladins (and similar features given to NPCs and monsters). Both Turn Undead and Control Undead involve Wisdom Saving Throws applied to Undead by people using their Holy (or Unholy) power against them.

So while I don't think this is a RAW decision that your DM made, I'm inclined to respect it in practice due to the similarities both in the source and mechanics of these features, and I might make similar rulings in my own games.

No future applications of Control Undead confer Advantage/Disadvantage

Directly related to the above is: when I recast control undead on an undead that I already own, does he have to make the saving throw again, or do I have an advantage somehow because I already own him?

The feature does not specify any such feature, so you do not gain any benefit under these circumstances. The only circumstance under which you would confer Disadvantage on their Saving Throw is if a different feature or spell is causing it.

You have two concepts here, so I've touched on both of them:

Control Undead vs Turn Undead

There is no reason for turn undead resistance to impact this. Turn undead is a cleric class feature. Playing an oathbreaker paladin, you are casting a completely different effect. While some paladins get turning effects, it seems that all turning effects have that specific word used in the description. Control undead, notably, does not. Now, I could see a DM using it for flavor, and I might be tempted to do the same, but there is nothing RAW that says this would be the case.

Control Dead Saving Throw

Nothing in the rules says that the undead creature would have disadvantage on this saving throw because they're already controlled. Charm effects generally have caveats about saving throws explicitly built in, but these caveats generally benefit the target. Furthermore, this could easily have the opposite effect of what you want to have happen, as control undead lasts until it is used again. You are, in essence, giving the undead a free saving throw, especially if you cast it earlier than the last possible second.