Don't loose hope - your feelings are very normal. I'm really sorry for your loss and know what you're going through.
First, family or not, I would have choice words for the sitter. Words like "neglectful" and "irresponsible" and "I'm expecting compensation for these losses those prior things caused". They don't have to be angry words - if the sitter knew what he was supposed to do and didn't do them and it cost the lives of your birds, it should cost him the lives of your birds - literally. As in what you paid for them or would pay to replace them... Just like if they broke a lamp or cracked your TV you'd expect them to pay to replace those things. This is a reasonable expectation - they broke it they replace it. Unless for whatever reason they didn't know any better? But it sounds like they did.
Second - Take the time you need to mourn and reinforce boundaries where you think is best but also know it's OK to start over if you want to or not as you see fit.
Third, definitely take your time to reinforce the fencing. Nobody should deride you trying to contain your animals and keep them safe through practical means and fencing is very practical. Even if you magically train your dogs perfectly tomorrow and they're never a threat again. What happens when a fox digs under? Or a weasel? Or a coyote? A stray dog? I lost my whole flock to one wobbly door and a particularly dedicated raccoon last year. Every bird in the coop, dead as a doornail overnight, bits all over the lawn, no dogs involved. Do what's best to keep your flock safe. Do whatever you see fit to upgrade your containment. Not to mention are other people going to show up at your door to train your dogs and watch your birds so you don't need fencing? No? Then do what's best for -your- situation!
Fourth, you can still trust your dogs, especially around children still. They don't view small animals like people. They're still the same dogs they were the day before, if maybe a bit more driven to hunt chickens now. And yes, you can theoretically train them out of attacking the chickens if you have enough time and patience. Not everyone has that kind of time and that's OK. Not every dog is so easy to break of their bad habits. I have a husky who was a chicken killer for five YEARS. Any small animal even so much as wandered into her eyesight in a place she was normally allowed to run loose and her eyes would glaze over and she'd drool and stand on the end of her leash panting and whining and fighting. She'd take out any hen that slipped the coop and and bunny whose latches weren't totally done up if I wasn't there to stop her and sometimes even if I was. I've lost a lot of animals to her over the years. Also she'd caught several wild animals over the years, squirrels, robins, you name it. And a stray cat. And while I got her to the point that she wouldn't actively chase if I was with her anymore and told her not to, and she'd totally and completely ignore the chickens even though they were just behind easily broken bird netting, she still chased and killed any that got out. The magic didn't really happen until she had something to hunt that wasn't the chickens. She's now allowed to hunt the groundhogs and rats on our property and because she has something she's allowed to hunt she totally ignores the hens. She can walk right up to a hen and ignore it because she's always hoping to hunt something better that won't get her in trouble. I don't think I'd trust her alone with the hens... But I'm no longer terrified when one slips out. And I think that makes all the difference.
At no point would I ever suggest that every dog can be totally trained out of hunting birds. That's a pipe dream. Some dogs will keep pushing no matter how many shock collars and prongs and rolled up newspapers or rotting dead chickens are applied (some common suggestions for teaching dogs not to hunt birds). There's a real chance that no amount of training will save your chickens if they're locked outside at night uncontained with the dogs. Dachshunds are bred to hunt small things. It's hard to get rid of hundred of years of inbred instinct. But almost every dog can be taught to defocus on the hunt and come when called. And that's big enough it could save your birds. Or it could make the dogs think twice about the effort needed to dig into a pen and decide it's not worth it. And that could be a big change for you.
So in summation, or tl;dr
- Yes it's OK to feel how you do. Your feelings are normal and healthy. And they will pass in time.
- I'd be very upset with the petsitter if they knew better and would ask for compensation, personally.
- Get chickens again if and when you feel ready for them.
- If you do get chickens - absolutely upgrade your fencing! It will be good for both your dogs and other predators!
- And yeah, train your dog, but it's OK and not unexpected if miracles don't happen and they don't 180 on the chickens with just a few sessions. Some dogs are unbreakable for that matter. That doesn't mean they're not still your beloved pets - it just means they're dogs.
- Despite all that there's probably progress you can make on training the dogs that will improve your scenario.