Changing the strings on an acoustic will make a big difference for you, for the better. Sound will improve dramatically, and you'll sound like a rock star overnight. Pretty much.
Regarding the muting of strings and whatnot, I spent a lot of time this past winter working on being sure that I was only hitting the bottom four strings on D and F chords, and the bottom five strings on A, C, etc. I practiced this by doing songs where I'd pick the top string, then strum down twice; gives it a good, solid, discernible bass note for each chord AND helped me get a better feel for where the strings were.
But when I'm just strumming away, I'll usually mute the 6th string on C, A, etc etc etc out of habit now. It works well in case I'm strumming hard and/or am hooked up to an amp.
Now, there was also mention of hitting the wrong strings and it sounding horrible. About two weeks ago I recorded myself and my sister (she's an amazing singer and uke player) on a pretty nice quality microphone. I'd never heard myself play before; I couldn't believe how good it sounded. In my head, I wasn't perfect and there were a few off-sounds, but on the recording I didn't notice it. And it was very good quality.
Ultimately though, my point is we can be perfectionists sometimes (which I think is good for improvement), but when the music your producing doesn't necessarily sound perfect it's important to remember it could still sound very very good to everyone who doesn't know exactly what you are doing.