First...what is causing the itching? Do you know? Is there a rash? A hot spot?
If not...I'd try this first, then if it continues or you see a rash...or if it gets worse, I would take her to the Vet.
Bathe her an an anti-microbial shampoo (kills yeast and bacteria). A good one is Mane N Tail ProTect (not regular Mane N Tail) You can get it at Farm Stores in the horse section.
Leave on for 10 mins as directed. Rinse well.
IF you want to use conditioner on the feathers...rear...tail...maybe chest...put it ONLY on the ends (in other words, don't POUR it on, use the palms of your hands), let sit for a few minutes and then rinse that area well with water. Don't put conditioner on the torso. Make SURE her body is totally rinsed. Then rinse again.
Then have a bucket of 1/3 apple cider vinegar and 2/3's water mixed up.
Pour over her (except for her head). Make sure you get it up under the armpits and on the stomach/chest/groin areas too. You can use a large sponge to slosh it up there.
Do NOT rinse out the vinegar and water. Let drip dry for about 10 mins, then towel dry and let sit for another 30 mins. Then dry with dryer if you wish.
Listerine is mainly alcohol so yes it is very drying. Vinegar is wonderful for the coat. It's a natural conditioner (they'll really shine)....and, it's also anti-microbial. So it really helps stop the itchies.
I'd go ahead and do the whole bath thing. Then you can have the vinegar and water in a spray bottle handy for spot application.