I definitely do look as bad as I think. I sing in a choir, and I have had someone in the audience come up to me afterwards and ask if I was OK! I need to take piles of tissues with me and constantly wipe my face, because the stage lights and/or the room gets hot. I also find that my eyes are very irritated, because I seem to react to the sweat getting into them. I have several anti-histamine eye drops that I've tried, but it doesn't stop the stinging and redness.
I'm not sure it's hot flushes, partly because it's gone on for a long time, and partly because it isn't just random like the hot flushes are. This comes with heat, humidity, or even minor exertion. And with any sort of more major exertion, like actual exercise, I sweat for several hours afterwards. I'm very unkeen on the summer!!
I tried the neck fans last year, and they were great while they worked, but every single one I ordered (about 5) stopped working within 1-5 days, and they all went back to amazon. If there was a more reliable brand that kept working for longer, I'd consider it. However my hair also got caught in it, which doesn't help.
Ikea has a cool gel pillow that I wondered about, but the reviews were mixed - some thought it helped, others didn't find it made a difference. I didn't try it in the end, as it's not particularly bad at night - I'm just generally hot then, but not the specific facial sweating.