Wow. This is the first time I've heard anyone express this concern.
It boggles my mind that anyone would care about being seen without a drink in their hand.
What consequences do you imagine will come from that?
If you are just walking around without a drink, then you're stalking/leering.
Of course it's much more nuanced than that - you can look like a lecherous creep stalking woman with a drink in your hand just as easily.
But that would be the premise: With a drink you don't need anything else to have a good time, you're already "doing something". Without, you're literally trying to fill a void.
Is this how you think of others, or do you imagine that's how they think of you?
For me, whether or not someone has a drink in their hand, if they're in a party or a club or a bar they're all doing basically the same thing: wandering, listening, looking to meet and talk to people.
I really don't get why holding a drink would change any of that.
Having some other activity involved (having a drink or otherwise) lowers the stakes when interacting with new people.
Most of human interactions is a facade of some sort. Smalltalk, icebreakers, most of early dating, even most professional conversations.
I know introverts and folks on the spectrum find this intolerable. But it's the foundations of human society.
You can hate it, but hating it will only make you angry. Instead, treat it like a game. There are many games you can play in the world to deal with something painful or frustrating.
If you're stuck trying to fall asleep, you pretend you're counting sheep jumping a fence.
If you're stuck in traffic, you try to guess which lane might be moving faster and make micro optimizations to get their faster.
If you're stuck looking for a life partner in a social setting, the game is act like you don't need one, that your life is fully complete without them, and that you don't care about any one specific given social interaction. Of course, the best way to play that game is to actually live that authentic life. But we all fail at hiding our emotions and true intentions in one way or another. So we play games to camouflage it like holding a drink and acting like that's all that we need to be engaged and interested.
I trick I learned long ago at house parties or BBQs is to drink one beer, then constantly re-fill it with water for the rest of the time. That way I have something to hold onto and a way to entertain myself where there is a lull in conversation or whatever.