Hi, Viv. I feel like I'm always jumping in to say yes! I have that symptom! but it's true! The cranial nerve symptoms are pretty predictable!
You may have a touch of facial weakness on that side even if not visible to the eye. I would definitely keep your doctor apprised, because they might want to offer you a course of steroids to relieve pressure on the nerves while you make up your mind about what to do. I found they sprang into action fairly quickly when facial symptoms presented.
I have realized in retrospect a few things that I noticed before diagnosis. I was never able to "wink" only my eye on the affected side, never, whereas I can singly wink the other side. I also bought a new light that I thought was making me dizzy the winter before I was diagnosed.
And I have never wanted people to stand or sit on my affected side if I could help it -- I always had to have people on my right side -- for decades before being diagnosed! I'm telling my friends from way back when, "See, there WAS a reason for that odd habit of mine!"
Point being, I don't think a drippy feeling on one side of your mouth is "normal" -- it is probably related. JMO. Keep us posted!