Have you tried Palm Muting with the heel of the thumb on your strumming hand? It is an extremely effective way to control your strings. With practice, you get to where you can roll the heel on and off the bridge and control your muting much like stepping on the gas, or stepping on the brakes. For example, if you want to mute your e and a strings you roll the heel onto those strings and strum away on the rest of the strings. I find this kind of Palm Muting is usually much easier to use than my freting fingers. Not to say that techique (of left hand muting) isn't useful, because it is. Palm Mutig is just to suggest another way to acheive the immediate goal keeping your e and a strings under control. I think that is would also be easier to learn, especially if you are still working on trying to switch chords without problems!

I would suggest you do a Guitar Tricks search for "Palm Muting". That will get you well on your way to getting your e and a strings to be cooperative with the c chord you want to play.