Does the undead controlled by the Oathbreaker paladin’s Control Undead ability get advantage/disadvantage on the saving throw?

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?