The timing sounds right for seasonal allergies in most locales too. I would also recommend treating it like allergies and seeing if it gets any better. I've had seasonal allergies my whole life and I have definitely become sore in my chest and abdomen from sneezing/coughing. If allergy treatments don't work, you can also try a humidifier. Like mentioned, stuffy noses are common during pregnancy, and the coughing could be from the air in your home being very dry. A humidifier might help.
Sounds like seasonal allergies. Stuffy noses are very common in pregnancy too. And lots of coughing will become painful even not pregnant.
Sinus Rinse works really well for stuffy and runny noses. It is dug free and entirely safe during pregnancy.
You can also try Zyrtec. It is the safest allergy medicine to use while pregnant and the most effective of all the overt the counter options. But do check with your doctor before taking it.