Welcome to HEALTHCARE MAGIC..
I have gone through your query and can understand your concerns..
As per your complain I would like to inform you that the hanging skin on either side of the midline that is not painful seems to be most probably a normal structure known as
Plica Fimbriata that is a normal fold of mucosa and is not a pathological structure..
It can be slightly sore due to
injury with some hard food, hot foods while at times concentrated mouthwash gargles an also make them sore..
Another reason can be infection in the floor of mouth..
You need not to worry as there seems to be nothing abnormal..
As of now you can do warm saline gargles and also suck ice to relieve discomfort..
Consult an Oral Physician if there is no improvement..
Hope this information helps..
Thanks and regards.
Dr.Honey Nandwani Arora.