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Thank you for this information.
When a dog is able to eat and swallow with no difficulty and is not showing signs of distress (such as pawing at the mouth or drooling), gagging does probably NOT indicate that they are choking on something or have something in their throat.
Similarly, if she is keen to eat and happy to play, nausea and a desire to vomit is also less likely (though not impossible).
More likely, this is an irritation /inflammation of the throat. We see this commonly in those with viral infection such as Kennel Cough.
Dogs with infections like this may have a gag / retch / dry cough intermittently for a few days that improves. They shouldn't have any other signs.
I would want to monitor her closely to ensure she remains well and doesn't develop any other symptoms such as vomiting, a loss of appetite etc.
I would also want to ensure she is up to date with her wormer and vaccines (I know given the current climate this has not always been easy).
How does this all sound?
No problem at all! Please do feel free to keep me updated. All the best with your new pup.