Anyone's 3 year old instantly lick their hands so they're sopping wet all the time?

My 3 year old picked up this habit about two weeks ago and we have no idea how to stop it. She constantly slobbers all over her hands. Her hands reek. She has sores on them from being wet. We've tried various moisturizers that do not taste good to deter her. No luck. We thought maybe her hands were dry but that's not it. If we ask her why she does it, she just shrugs and says "Because". We've asked her to stop, we talk about how gross it is, and nothing. Other kids don't want to play with her because of this but she won't stop.

She's supposed to see the doctor next Friday but I'm going insane because that's more than a week away.

It's frustrating because her hands are ALWAYS, ALWAYS wet. She can't open door knobs because her hands are too wet. She can't colour because her hands are too wet. We've tried redirecting her, giving her things to chew but she's not chewing on her hands. She literally fills her mouth with saliva and drools all over her hands repeatedly all day.

Her daycare teacher has never seen anything like it. We tried the nail polish that tastes bad and that did nothing. She's not chewing on her hands, she's putting them in her mouth to drool on them.

Please tell me you've solved a similar problem because I am at my wits end. We've tried gloves but that doesn't stop her either.