New puppies vs existing adult dog...HELP!!!

I have had a female Jack Russell/Beagle mix for about 3 years now her name is Bella. My husband and kiddos were desperate to get another puppy and we ended up with 2. They are both male teacup maltese/jack russell mix (brothers). Their names are shadow and ghost (kiddos named them).

Bella has always taken to me and been protective or possessive of me. But has never bitten at anyone. When we brought the puppies home, she sniffed at them and it seemed ok. On Thursday Shadow was going to drink water while Bella was eating and Bella bit Shadow on the face and his eyelid was bleeding.

Today, both puppies were on my lap and Bella tried to bit Ghost for no reason at all.

How do I fix this? Is there a chance to fix it or is Bella not retrainable? I am exhausted trying to keep them separated so as to avoid a disaster.

All dogs are trainable, she's no doubt feeling pushed aside, make it plain her behaviour is unacceptable but stop giving the puppies privileges.
You'll probably have more issues with 2 male terriers as they mature so best start training them all early as to what the boundaries are.

How do I correct the adult dogs behavior? I am incredibly hesitant to even have her around them because she is so antsy around them. She shakes, growls, barks. I am afraid something very bad will happen.

What commands does your dog know?
Tbh keeping her away will make matters worse unless tou are spending quality one to one time with her.

You don`t, you teach the pups to respect her , would`nt you snap if you were enjoying a drink or food and some young upstart came and tried to pinch it from you,

You are going to have to make sure when she eats /drinks.toys, the pups leave her alone, feed her where they cant get to her.

If you want to give the pups cuddles do it when she is not in the room, and vise versa, she should be having the special privileges , the other two don`t know any different so should fit in with your / her routine.

I would think you are going to have your hands full enough with two siblings, so best to set down some rules now, but its not fair on the 3 yr old to have to fit in with them, they need to fit in around her.

When we feed her, we have always set the food down and told her to sit then wait until we said ok to eat. When we play with a ball she is good with being told to drop it or put it down. Never has been good with telling her to come back when she bolts out of the house before we can hook the leash. It really is not just when she eats that she gets upset. If the puppies even walk around the room when she is on the couch she starts to growl.