Because her symptoms are intermittent, and she's now comfortably sleeping, it is unlikely that anything is truly stuck so don't be too worried about looking in her mouth or down her throat.
It can be hard to tell the difference but this could actually be a cough.
A cough simply means she is experiencing airway irritation.
A cough can be related to heart disease, including heartworm, lung disease secondary to a viral, bacterial, fungal or parasitic infection or tracheal irritation due to a bacterial, viral or fungal infection.
In a young dog infectious disease is by far the most common cause. Of those kennel cough or infectious tracheobronchitis is one of the most common causes of a gagging cough. If she has been to a groomer, dog park or boarding facility or gets out for walks then she can be exposed to viral infections, so kennel cough would be high on my list of possible problems.
Even if she has been vaccinated we cannot vaccinate against all of the agents that cause kennel cough and some dogs will still get sick even with the vaccine. The good news is that dogs that are vaccinated usually don't get as sick as dogs that haven't been vaccinated.
As long as she is still able to eat and drinking well, isn't lethargic, isn't running a fever (rectal temperature greater than 103F), and she doesn't have a green or yellow nasal or eye discharge then you can try taking a watch and see approach and using an over the counter cough suppressant if her gagging cough gets to the point that she cannot sleep comfortably.
Make sure she gets plenty of fluids, lots of rest as the more active she is then more she will cough/gag. Add warm water or low salt chicken broth to her food. The better hydrated she is the better she will feel and the thinner her respiratory secretions will be.
You can take her into the bathroom with you when you take a hot bath or shower so she can breathe the steam which is soothing to her airways and thins mucous, making it easier for her to breathe. You can also use a humidifier in the room she sleeps in,
Her cough should slowly resolve over the next 2 to 3 weeks. Most of the time these dogs feel pretty well so it's a matter of keeping them quiet and suppressing the cough if we can.
If the cough is very bad you can use human cough/cold medications with Guaifenesin (an expectorant) and Dextromethorphan (a cough suppressant) only such as Benylin or Robitussin DM can help. Give 1/2 teaspoon per 10 to 15 pounds every 6 to 8 hours.
If she is running a fever this is almost certainly a contagious infection and if she is running a fever and is lethargic then an examination is definitely warranted as pneumonia may have developed.
Other reasons for gagging include gastroesophageal reflux. That can be due to changes in food or treats, overeating, or eating something outside or around the house causing irritation.
At home to help with reflux you can give either:
1) Pepcid ac (famotidine) at a dose of ¼ of a 10mg tablet per 5 to 10 pounds of body weight every 12 hours
2) Prilosec (omeprazole) at a dose of ¼ of a 20mg tablet per 10 to 20 pounds of body weight every 24 hours
These are acid reducers and should help if this is reflux. They are quite safe and can be used for several days if necessary.
Best of luck with your girl, please let me know if you have any further questions.