I'm sorry to hear about Billie's frequent vomiting. Cats normally don't vomit that much from hairballs.
A sudden diet change can exacerbate vomiting too.
Has anything else changed about Billie?
Is she hungrier than she used to be and gulping her food without it chewing well?
Is she losing weight?
Is she drinking more water and urinating more?
Dry food is likely more of a problem than canned as it is easier to eat fast and it expands once it hits the stomach juices, making the cat feel over full and thus vomit.
But if this is new behavior for her perhaps there's more behind it. And if she's losing weight or drinking more water there almost certainly is another problem.
She may be hungrier because she has an overactive thyroid gland due to a thyroid gland tumor. These cats have an increased metabolic rate and must eat more to maintain their bodyweight, yet their stomach is the same size. She may be hungrier because she is diabetic and isn't able to use the calories she is taking in. These cats generally are drinking more and urinating more as well.
It's also possible that she has developed pancreatitis (an inflammation of the pancreas) or inflammatory bowel disease and both can lead to increased bouts of vomiting.
I now that you would like to try some things at home but I recommend that you start with a physical examination and some blood tests at your veterinarian. Hyperthyroidism and diabetes are controllable with medication, and pancreatitis and inflammatory bowel disease can be helped by dietary changes.
In the meantime until you can have her checked out I would feed her canned only, that is less likely to cause vomiting in many cats. You can also give her pepcid-ac (famotidine) at a dose of 1/4 of a 10mg tablet per 5 to 10 pound cat every 12 hours. This antacid medication may help settle her stomach a bit.
I recommend offering a bland diet of 2/3 boiled minced white skinless chicken and 1/3 boiled white rice mixed with some low salt chicken broth to make it more palatable and get additional fluids in, which will support her organs.
You can use hairball preparations like Laxatone or Petramalt, although I'm not convinced they will help.
If she seems to improve on the bland diet mix both Royal Canin & Science Diet brand cat food companies make a sensitive stomach formula for cats you can try transitioning her to. Make sure to transition her very slowly, adding a little more of the new diet to each meal. It should take a week to 10 days to transition her slowly.
Please let me know if you have any further questions.