cleaning up my chords

... I hear a lot of excess string ringing in between chord changes and some other string noise that muddy up the songs I play.

Sounds like you are hitting strings that are not part of the chord. For instance, when you play an open A - are you hitting the 6th string E? This would cause the chord to sound muddy or as you said, add other string noise. When I first started, often times I would practice a chord and tried to be very carefull to make sure I hit all of the correct strings. I then found myself hitting additional strings when trying to play the chords faster in a rhythm.
Are you using an excess of downstrokes or upstrokes? This may cause unwanted string noise as well.

Muting can be achieved in a few ways. You can mute strings by lightly touching them with your fretting hand. Using the example above, when I play an A, I will often mute the E string with my thumb by hooking it slightly over the neck of the guitar.

Palm muting would require you to place your strumming hand at the bridge and apply light pressure to the strings you wish to mute. It takes practice to develop the right touch. I would suggest you start by playing 1 chord and try to mute it. Repeat the process until you get the sound your looking for. After you develop a feel for it, you'll be able to try different techniques like playing an open chord and letting it ring through partially or half way through.

I do hit extra strings when I play chords but do my very best to do what you're saying and mute the E string on an open A or C with my thumb or with the tip of my index finger on an Am shape barre. One difficult chord is open D, I can mute E string but not A, so I often hear the open A string in my D.

I'm guessing most of my issues though are the open strings in a chord that may ring through to the next chord that when I try lightly touching or lifting my fretting hand won't mute. Is it pretty common to just let them ring through?

For instance, when I listen to an original recording of a song that I play along to, I will hear a lot of open strings ringing in my chords that I may not hear in the original. To me this sounds noisy & less clean than the original. I know that I can't be the guy on the guitar in the recording, but I think that there's a technique he/she must be using to sound cleaner than I am (aside from not hitting extra stings)

I think this comes down to practice and repetition. If you have the means, record yourself now, practice this for a week and record yourself again. I think you'll be amazed at the difference.

Keep in mind that in professional studio recordings, people did things after MANY rehearsals and MANY, MANY takes. It should sound clean and crisp.

adjust the action on your guitar.... that will take out some buzz... also maybe use some higher guage strings ( it worked for me) also know that whenever playing open chords with solid state distortion turned on the distortion will cause the notes to clash against each other not sounding very clear....

hope that helps

I get what you mean with distortion, but what is solid state distortion?

Never really qualified this....
When you say you need to clean up your cchords...what are you playing? Electric or acoustic? (I assume electric)
If that's the case...how much gain or distortion are you using?