Hey jammer!
First and foremost, it's going to take "some amount of time" to get your hands and fingers to "know" where you want them to go without thinking about it. Right now your brain is telling your hands OK, I'm playing a G chord and now I want to play a C chord. Your hands are saying, OK, no problem. But just where the heck is a C chord and how to I get there. Your brain is signaling your hand but your hand can't keep up with your brain just yet. I'd say in about 3 to 4 weeks, you'll be comfortable in switching between the most common major chords. IF you practice!
You have to build up muscle memory. It's like when you first started writing cursive. Your pen or pencil was all over the place. But with practice you soon learned to stay inside the lines. Your hands and fingers "memorized" how to make the curly-que E.
Same with guitar. You have to spend some time practicing chord changes. Its just like everything else you have learned to do. By practice.
Yep, it's that P word. Practice!
And do not spend time worrying about whether your fingers are to fat. They aren't. There are plenty of chunky fingers out there making the tunes. It's a matter of learning where to place your finger tips and how to "curl" or "arch" your fingers so not to interfere with the other strings. Unless you have an extremely narrow neck, 1.75 or smaller, you'll be fine.
**
LC