BBb tends to have a little fuzz in the sound compared to CC.
A fuzz or airy quality in your tone will come from a bad buzz quality in the mouthpiece or a bad mouthpiece fit/gap in the horn and can exist equally in any and all horns. Unless you are dealing with some sort of mechanical problem, most of the time it, is from a bad match between the horn, the mouthpiece and your embouchure or an immature embouchure. This is why you find players on mouthpiece searches all the time on this site.
Having played BBb, CC and Eb over the years I'd would recommend staying away from trying to transition from the BBb into the CC. The only advantage I know of in making the transition is in the ease of fingering runs when you are stuck playing in those notorious sharp key keys so often encountered in an orchestral setting. You can find great horns with fantastic sound characteristics that match or beat the CC, in both Eb and BBb. A large 5/4 Eb would probably be the most versatile horn you could buy. You will be able to drop down into the deep full BBb characteristics of a tuba by pressing the 4th valve and still have the light characteristic need for solo or small groups the rest of the time.
If you are playing orchestral music all the time it makes sense to have a CC, but I have never found any advantage in having taken up the CC while playing American music in an Army Band, or in any American Wind Ensemble or in British style brass bands or in any Dixieland Bands. In the teaching environment you will have little need for the CC. Taking up the CC while I was at the Military School of Music was a massive waste of my time and energy. Unless you intend to be playing at the professional symphony level, except for some preparatory collegiate playing for the symphony scene, you won't have any need to ever be playing the CC.
As a side note… in every audition I have ever been through, I was never asked what equipment I was using. They were only interested in my technical ability and quality of my sound that was actually reaching them in their seats.