I am very sorry to hear about your pup Claire's restless behavior, licking the floor, eating grass, panting, coughing and gagging.
Dogs with nausea or gastroesophageal acid reflux often eat or lick grass or foreign material to make themselves vomit. They will often swallow forcefully repeatedly, gag, and lick things trying to soothe their nausea and acid burn. If they can rid themselves of whatever is irritating their stomach that way then all is good.
But when they are gagging repeatedly and continue to be uncomfortable then we need to stop them from eating any more grass. It becomes a vicious cycle where the more they vomit the more their stomach acid burns their esophagus and the worse they feel.
Possible causes of nausea causing symptoms like ers include a sudden change in diet (food or treats), dietary sensitivities or allergies, or eating things that they should not like too many fatty table scraps or garbage, bones etc. Addison's which is a poorly functioning adrenal gland is another possibility for waxing and waning vomiting and nausea. Metabolic organ failure (kidney or liver disease), pancreatitis, inflammatory bowel disease or even infiltrative cancers are possible causes as well at her age.
Do you know if Claire was fed anything different, or got into anything?
You can give her an acid reducer to try and settle her stomach. Either:
1) Pepcid-ac (famotidine) at a dose of one 10mg tablet per 20 to 40 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at one a dose of one 20mg tablet per 40 to 80 pounds of body weight every 24 hours.
These will reduce stomach acid and could help settle her stomach. These can be used for several days if necessary as they are very safe.
I'd also pick up her food and water for now. A couple hours after the acid reducer is given you can offer small amounts of water or ice cubes to lick. She's likely thirsty but we need to settle her stomach first.
No food for 12-24 hours. Small amounts of water only.
After her food fast then start a bland diet of 1/3 boiled, lean hamburger (or boiled, white, skinless chicken) and 2/3 boiled white rice. Give small meals several times a day. Feed the bland diet for several days, then if things are going well start mixing in her regular diet and slowly convert her back.
If her nausea continues then she may need injectable medication from a veterinarian to get her stomach upset under control.
So watch her for continued vomiting even with the acid reducers, blood in her stool or vomit or a fever (more than 103.5F rectally), a tense painful belly or lack of appetite after the acid reducer and her food fast. If any of those occur it is time to seek hands on veterinary care.
It would also be a good idea to have blood tests done as well to make sure there isn't an underlying metabolic problem like pancreatitis or Addison's disease present.
Please let me know if you have any further questions.