Does Stress Cause Dry Cough?
Less likely to be stress, more likely to be allergy or from an irritant.
Detailed Answer:
Hello and welcome,
Stress doesn't usually cause coughing fits or coughing with sneezing. As this started after you moved to a new house, it could be something in the house or environment around the house that is causing either an irritant or allergic type of reaction.
About the GERD, if you are bending over a lot to pack/unpack, this could exacerbate the GERD which could contribute to the cough (but would be unlikely to make you sneeze).
So what to do? If the GERD seems worse, I'd recommend taking over the counter Zantac 150 mg twice a day for several weeks. That way you will know if making the GERD better makes the coughing better.
But as sneezing is also involved, there may be some sort of irritant you are being exposed to. I think it would be best for you to see your doctor who can listen to your lungs (and possibly do a chest X-ray) to see if he or she can sort this out.
If you are unpacking, consider wearing a good mask much as the 3M N95 mask which you can get at a hardware store. You may need to do some sleuthing to see if there is something in the new house (or general environment) that you might be sensitive to.
I hope this information helps.