Serious Separation Anxiety

So here's of the story of my parent's dogs.

There is this pair of two year old female mutts that have been together all of their lives, constantly side by side and they have never been apart. They were the only puppies in a litter of two. They haven't had any training at all, not even leash trained. They were not messed with as puppies. Recently I decided to "train" them as my parents are throwing around the idea rehoming one of them.

The brown one is clearly the dominant one and is aggressive about food and her sister. We must feed them in different sides of the kennel, or else the other one won't get to eat. She is more interested in humans and seeks attention. She is hyper, however settles down when someone starts petting her (a trait that her mom has).

The white one is the more shy dog. She's a little harder to get to. She will cower a lot to anything. She'll often move away from you or just lay down and cower. A very submissive dog. She hasn't been abused or had any bad experiences with humans, just a very anxious dog. She is more quiet than her sister. During feeding time, her sister will come up to the kennel door while she waits in the back of the kennel for her share. She's not very fond of people and prefers not to be near them. She won't even sniff your hand.

Yesterday, I bought a collar for both of them and decided to take them out to see how they did.

Getting the collars on was easy.

They both did fine together and didn't completely choke themselves or resist the leash, but once separated, they both got very anxious.

My dad took the white one and I took the brown one. As soon as the white one began walking in the other direction, the brown dog went crazy, as well as the white one.

We thought maybe if we moved them out of sight from each other that they'd calm down, however it was just a lot of yapping and howling for the other. I'm very patient when it comes to animals but I didn't really know what to do, so I just sat there and waited and tried to comfort the dog. The dog was not interested me at all. No matter the amount of petting and rubbing, even treats, she was way too anxious. Tried to take her on a walk to other places, but she would be hesitant to go far from the property and would keep looking around or howling for her sister.

Afterwards, we switched dogs and I handled the white one. This time, we kept them a very short distance apart, about 4 feet away from each other and fed them their dinner. Even then, they were still very anxious and would constantly try to get to one another, barking and yapping and circling. The brown one ate, however the white one was skeptical of the food. Instead of jumping around trying to get to her sister, she'd just lay down and stare or occasionally bark.

Then we tried to take them out of sight from each other again. This time I kept the white one with me. She was yappy, but not as bad. She's more of a whiner. She would lay down, whine, bark a couple of times, stand up, look, repeat. I did the same thing with the brown dog. I tried to comfort her physically, with food, with a walk. She really couldn't care less if I was touching her or not. Usually she would cower if I went to pet her head, but she didn't even care, her eyes would just stare right ahead. Was not interested in the food at all. During the walk, she was willing to go out for a bit, then immediately turned around and started barking for her sister. Also, while she's anxious, she'll resist walking. She'll just plop down the floor and lay there. Doesn't seem to understand coaxing, she does not want to come to me but gets the idea of not dragging me on the leash. Not long after, I put them together on a leash, side by side, and they did a lot better and surprisingly never got tangled.

I've never dealt with dogs before so I'm not sure what to do or how to handle it. I do work with horses and a common way to train horses is by pressure and release, however i'm not educated on dog training, especially when they are very emotionally attached to each other. If they aren't so distressed from being separated so that I can work on basic obedience, that is fine with me. I'm not really sure where to start.

Advice on anything and everything is very helpful. Critism is welcome as well so I do not repeat mistakes. Thank you for your time and in advance!