Welcome to our forum GroverCat. Unless you have a severe case of Primary Sjogren's, meaning it's the first and most prevalent of your autoimmune diseases, the most important thing to do is treat the symptoms. If I'm reading your post correctly, Hashimoto's is your primary autoimmune disease. Your dental issues could be Sjogren's because dry mouth causes dental problems. Most folks have greatly reduced saliva before they even feel dry mouth. You can be tested for Sjogren's antibodies, but there are folks who have the antibodies yet never develop symptoms. So, test or don't test, one still treats the symptoms.
Since you are having lots of dental problems you may consider starting the OTC products for the mouth. Use a Waterpik for gum issues, Biotene toothpaste, Biotene Oral Balance Gel at night, organic lollipops or gum with xylitol, and sips of water to keep your mouth moist through the day. Lastly a mouthwash without alcohol which restores enamel such as ACT.
GERD can be associated with Sjogren's. Your primary care can prescribe medication for GERD or you can buy products OTC. I hope this helps answer your questions. Love, Butterflake