I just started learning ("learning") how to play guitar. I've been going through the basic chords and scales, and I've noticed something that I'm not sure I can change.

When I'm doing chords, it's seemingly impossible for my fingers to NOT be touching some other strings and muting them. I've tried putting my thumb directly behind the fretboard, perpendicular to the strings, on top of the fretboard - no matter what I do, my fingers are muffling another string. On G major, it's my index finger muffling the A string, for example. Then when I try to curl my fingers more to clear the other strings, my third finger moves off of the high E string. And with other chords, it's the top of my palm muting the high E.

Do I simply not have the fingers to play guitar? This seems to me something that can't be fixed, especially when the only way I can play with no strings being muffled is to curl my wrist into the most god-awful awkwardest of all positions.
You are new, I was the same way (back in the days when a F power chord was nearly impossible lol). You will get the hang of it, just keep working at it!
I know I'm new and everything, but it just seems like my hand doesn't work the way it needs to at all.
It'll be fine, all hands suck when you first start, chords will be your bane. Once your calluses start building up and you get more strength and dexterity in your hands you'll be able to play chords much easier. Over time your hands will get used to the instrument, don't get worked up about it mate.
Have you tried tampax?

Work.hard, play hard
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It's the same for everybody. Even the best players in the world will occasionally mute an adjacent string while fretting a chord. Just keep at it and your accuracy will get better and better.
Yeah, when I started I played everything with my middle finger (I actually did, it felt more natural than using my index).
That happens to every beginner! You are not completely used to the guitar yet and with time your playing will get better and cleaner. Make sure you're holding the guitar properly. A lot of people have issues with muted strings because they are not positioning the guitar properly. Just look it up on youtube there are several beginner lessons which will cover this.

Good Luck