Old time Andrews Sisters like backing vocal sound?????

Fantastic question

now I am just guessing here, but you should already have recorded a tight 3-part harmony in a punishing falsetto - (or used the vocal transformer to imitate a root, major third and fifth chord)

so maybe now you should consider what sort of equipment was common at the time - ie 'lo fi'.

Just a theory, but open up the eq and drop everything down, then pick a few tight frequencies and ram them up nice and high around the vox levels

Just thinking about tinny little radios with tiny little speakers

add a smidge of distortion and how close are you getting?

That "sound" is in their technique in singing. It's very hard to do, but it can be done. Find a trio of good female singers, sit and listen to their tracks over and over again, the try to reproduce the harmony and playful style of singing. They were very tight--vocally, so it will take a lot of rehearsal.
You might want to also rent Spielberg's 1941 and the following classic films...

Road to Rio
Her Lucky Night
Follow the Boys
Hollywood Canteen
Moonlight and Cactus
Swingtime Johnny
Always a Bridesmaid
How's About It?
Give Out Sisters
Private Buckaroo
What's Cookin'?
In the Navy
Buck Privates
Hold that Ghost
Argentine Nights

I'd just add to the already excellent suggestions that the Andrews sisters' signature arrangements used a lot of 1st and 2nd inversion chords, and were "rhythmically perky."

The most important thing in replicating their sound is the arrangement and performance. The recording technique while also important, is not as important as arrangement and performance.

Ahhh, I missed the emphasis on "SOUND". Then yes, the EQ setting in the way to go. A better way to ask the question would be "that 40s radio" sound. When you added the Andrew Sisters, that too was a "sound" of a different nature--aka "style" or "technique".