Fear not. You can learn to play very well with fat fingers! I know from personal experience.
I am cursed with short fat fingers and when first learning to play chords I experienced the same thing you are experiencing. In fact I believe all beginning guitarist experience this to a certain degree.
As others have mentioned, you might try other guitars with wider string spacing which will allow more room between strings.
I have guitars which are more comfortable for me to play than others. But with practice, I have developed the ability to cleanly fret chords on guitars with the narrowest of string spacing (in spite of my fat fingers).
As you probably know, when possible, your angle of attack on the strings should be with your fingertip perpendicular to the fretboard. Obviously to do this your fingernails must be very closely trimmed.
Not every chord can be played with a perpendicular finger placement on every fretted string. On some chords you may need to find an alternate finger placement that works. On some chords you might have to slant one or more fingers and contact the string with the side of your fingertip.
In addition to finger placement and angle of attack, the other thing you might need to adjust is your thumb position on the back of the neck. I find that to cleanly play certain chords without muting strings I am constantly readjusting my thumb position on the back of the neck. Every chord formation gets its own thumb position. It will eventually become very intuitive to include the thumb position for certain chords as part of the finger position for that formation.
For example, I know when I play an open E major chord I need to shift my thumb to the treble side of the back of the neck to get my fingers oriented perpendicular to the fretboard. So while my fingers are forming the correct position in mid air - my wrist is rotating so my thumb can slide closer to the treble side of the neck. When I play an open D major chord, my thumb is pointed straight up at the ceiling and the tip of my thumb is not even on the back of the neck. The C chord finds my thumb more parallel with the neck of the guitar.
In essence, every chord formation involves getting your fingers in the correct position AND a concurrent adjustment in your wrist orientation and thumb placement and orientation on the back of the neck. So it's not just learning the finger positioning. You will eventually internalize the subconscious adjustment of every part of your wrist and hand for each chord. It's not as difficult as it sounds and with practice - becomes automatic.
Try different thumb positions and thumb orientations and different wrist positions with the chords that are giving you trouble. You may find that a shift in thumb position or or wrist angle might make it easier to get your fingers where they need to be. Other guitarist may be able to show you what they do on certain chords and a teacher might help you with alternate positioning. But everyone's anatomy is different so ultimately you must discover what works best for you.
Take it one three chord progression at a time so you don't get overwhelmed. With practice (and a guitar suited to your personal comfort) you will eventually be playing all chords cleanly. Enjoy the journey!