A great way to learn finger independence for your right hand is to practice different patterns, so I'm going to show you some different patterns.
Let's start off with the Travis picking pattern with your thumb again. I'm just going to do it on a C chord, just your standard open C chord. And I'm going to start with the fifth string with my thumb and move up to the fourth string. You'll notice that my left hand finger's coming down on the sixth string. I have to hit that G note down here on the sixth string, and then back to the fourth string with my right hand. In a nutshell that's what the sort of Chet Atkins, Merle Travis picking is all about. That's a really great exercise for your thumb to keep that happening, to get that sound.
Another good thing to practice is patterns as I mentioned before. So, what we're going to do is you can pick any chord you want, let's just stick with that same C chord, and I'm going to do a finger roll of thumb, index, middle, ring finger. I don't advocate practicing in front the TV, but maybe this is something that you actually could practice where you're not having to think about it so much and it just becomes automatic.
You can do the reverse of that. Start with your thumb, ring, middle, index. You can practice moving your thumb around. Try to come up with different combinations