The prescription label will tell you how to use your mouthwash and if you can swallow it or not. Usually a topical treatment for mouth sores, it's likely OK if you swallow some.

The impact of mouth-taping in mouth-breathers with mild obstructive sleep apnea: A preliminary study. Healthcare . Vol 10, September 13, 2022, p. 1755. doi: 10.3390/healthcare10091755.