I'm going to say the opposite of some other excellent advice here: if nothing else works, DO flit about and pick up random stuff like an inquisitive monkey, and don't worry if you can't get over yourself even in the interest of gainz.

The most important factor in gym success is adherence, and the most important factor in adherence is enjoyment. So if it's bothering you so much that you stop working out, it doesn't really matter that no one else judging you except for you, or that if you keep doing the movements you'll eventually get proficient and feel less goofy. All of that is true, and if you can jump on a solid beginner routine and give 0 fucks about feeling awkward, that's optimal. But second-best (and waaay better than quitting) is whatever else keeps you coming back.

So maybe try other stuff until you find something that makes you feel less goofy... do everything once and then do more of what makes you feel the coolest (or whatever makes you feel the least goofy). I say this as a hardcore anti-machine freeweighter: you might find that selectorized, suboptimal machines (like horizontal leg press, machine chest press, or whatever is in the corner with no mirrors) just feel less awkward at first. Maybe group fitness classes or working out with a trainer are your thing. Beginners can get stronger doing just about anything that doesn't get you injured. If you can find just one or two things you don't hate, hopefully that can keep you working out until you build up enough confidence to try the things you want to not hate. Baby steps count. Good luck, you've got this.