I accidentally swallowed a bit of chlorhexidine gluconate while gargling with it for a sore throat about an hour ago. Should I be worried?

Don't worry. The small amount that you swallowed while rinsing is not enough to cause any problem. Do not worry about this.
No you should be . Fine. I wouldn't worry about it. Use it as specified henceforth - you should do fine. Hope this helps.