I've done the same thing, although not with Corgis. I had a much older dog and ended up taking in an abandoned puppy, which was very trying on my existing dog. They had different personalities since the new one was a puppy, and the older dog took a few weeks to get used to sharing toys. Walking was a nightmare for a while (after the puppy learned what a leash is about).
There are also a few things you can do with the walking problem. Combined with the harnesses to make the pulling manageable, I recommend "leash training." You bring treats and praise him when he's walking by your side with slack on the leash for a certain amount of time (or distance, steps, etc). If he starts to pull or freak out, stop and reign him in, making him sit or stand by you until he's calm again. Then keep walking, rinse, repeat. It's soooo hard to start with, but pays off with a good (or at least improved) leash walker. It used to take me the best part of an hour to go two short blocks until my puppy got to the point that I found acceptable. Then there is walking them together. You will need to have someone else on the walk with you, walking the other dog. Walk together and repeat the leash training thing. New pup will pull more because he wants to keep up or play. If this happens, reign him in until calm. If he won't calm down, turn and walk in the opposite direction for a while. Let the other dog keep walking until just out of eyesight or until they're a decent distance apart. Rinse, repeat. Again it takes a long time, but ends up worth it when you can walk both together.
Crates are a great idea to solve food problems. I, personally, simply trained my dogs to wait until I gave a command so they could eat. I started out with the bowls a good distance apart and DH watching the older one. If anyone got possessive or so much as moved toward the wrong bowl, both got taken away until everyone was calm. Rinse repeat. Again, worth it in the end since they could eat nearly side by side after that.
I also always made sure that DH and I were dominant, to help ensure our will would be followed. Anything that caused a fight was taken away immediately, and no one got it for at least an hour. It helps immensely to become familiar with warning signs that the dogs will fight soon, or that they are tense. Stopping things before they escalate is important, and much easier to deal with than an actual fight.
For barking... I let my dogs bark, but they're not compulsive. They bark at anyone coming up the driveway, when another dog passes too close to the house, and if someone is on the lawn. I find this acceptable, but do call them down if it goes on for too long. I've heard that teaching them to bark on command helps, since, if you can command them to bark, it follows that you can teach them when not to bark. Or at least get them to wait for the next command to do so. Reward-based training is always better than punishment, as punishment requires escalation. Remember that as little as one second between behavior and reward can impact how well your training goes.
My vet told me that there was no real point in buying weight loss food. Instead, I was told to buy frozen green beans and start off cutting my current amount of food by 25%. After that didn't work, I cut it again. Fill in the difference with the plain green beans to make them feel more full and give them the impression they are eating a lot. It has never interfered with my dog's digestion, and now she likes them. It's taken a couple months but she is visibly slimming down. I'm grateful for that since fast weight loss isn't healthy, and since I have a Sam's card the green bean solution isn't particularly expensive. If your dog is having digestion issues, most food companies have policies that state you can return the food if the bag is more than half full for a refund. Check per brand before you do this, but it could save you money and an upset puppy tummy. Make sure that you titrate over the food when you change, though. Changing too fast can cause an upset tummy as well.
Things sound doable for you. It's just a lot of work and time for a while, but getting to watch them get along and having two faces to greet you every day will be worth it!