A friend of mine had a teacher that used to say, "Sometimes you damp with the right hand, other times your left, and sometimes you use your nose if that's what's available." He played a 10 string so damping was somewhat more difficult. He was kidding about the nose (I think) but his point was that you damp with what you can...
I would say as a general rule right hand is preferable but sometimes you'll need to use the left. Sometimes tips and sometimes the edge, sometimes simply releasing pressure on string held down with left hand finger (although not necessarily moving the finger off the string), sometimes the right hand thumb, sometimes you can use a rest stroke to dampen adjacent strings... This is something you'll develop over time. The main thing I would avoid is changing from proper hand position to use it for damping like flattening left hand fingers (playing with pad) to accommodate muting a note. This is really typical in electric playing (example: muting at bridge by resting pick hand on the saddle, sometimes reaching over top of neck with left thumb, etc.) but with classical it's best to develop good technique and optimum hand positions and let the damping fall into place from there (rather than altering angles/positions to dampen notes and having to adjust your technique later).
Scales are a good way to develop good damping techniques... Some say always practice scales using rest stroke, but I find that practicing them with free strokes and using thumb to dampen is useful too. Have your teacher show you how to properly use scales to strengthen overall technique. Also planting is another great way to dampen... Ask your teacher to explain how that works. If you learn several methods to dampen notes you will then also know when you need to dampen by playing on pads (i.e. no better alternative available).
One thing I'd add. I'd say the fact that you are aware of the need to dampen notes and not just let them sustain means you are way ahead of the game and doing better than a lot of beginners. Truly one of the most important thing to getting melodies and bass lines to sound clear and independent of one another. I overlooked that one for years and could never understand why my bass lines (sometimes melodies) just seemed to get lost in the mix.
One Ring to rule them all, One Ring to find them, One Ring to bring them all, and in the darkness bind them.