These two evils should be the first thing to look at if you're having problems sustaining notes or are damaging your lip, incidentally. Now, gradually, let your lower lip apply less pressure on the reed until the note begins to drop. Then, tighten your lip again to sharpen the note and get your C back. It's harder on the lower notes, and easiest in the middle register. You will need to supply more air at the bottom of the slur.
Vibrato is a useful technique.
If you can master slurs, then vibrato is the next important effect. A point worth remembering is that when you're on stage or using a microphone, you need more vibrato for the audience to get your tone. The net result is a flat note. Ideally, you need to be able to sharpen your note as well as flatten it. It makes you flat.