I've got to back up and make string damping a priority. Do you think it's a good idea to practice different scales and use my fretting hand while strumming all the strings? How did you develop this technique and keep it fun?
I always kept my learning fun by playing music I love. You have to damp almost all the time and in almost every playing style so you might as well just practice damping on whatever material you normally work on. If you want to get specific on damping/muting work, find stuff you like that has it. Learning to play a single note phrase while strumming all the strings is definitely a good exercise and one that won't be boring if you like the phrase in question.
For practicing left hand damping, that technique is one which will work, although it won't necessarily help you learn accuracy. You could also try practicing fast damped arpeggios - listen to some Al di Meola for great examples.
Also remember right hand damping has it's place as well, either using the edge of your hand (at bridge for light damping or further down the string for heavier, more Metallica-esque damping) or the remaining fingers on your right hand to damp specific strings.
There comes a point with my pupils when I say 'go home and play solos very loudly with distortion'. This meant that there had to be a lot of damping going on, otherwise string noise became more unbearable than the playing itself!. Parents got to understand, and the excuse was enough to warrant extra practice, working out personal ways of damping, using palm of R.H., fingers of L.H., not taking fretting fingers off strings completely, etc.
Your idea of strumming all strings to produce just an individual note, from a scale or whatever is not really a satisfactory way to do it.
I've found that when I need to work on this skill, I use a sixteenth note repetition doing 8 beats with a string wide open and then 8 beats with the string muted. The best way for doing this, to me, is with a palm mute technique. Set a metronome to whatever pace is comfortable to you with 16th notes and move from the low E to the high e, a string at a time.
Once you're good with the single string muting, you can also do some chord muting with your left hand. A lot of drop-d style songs incorporate this, so you may want to drop that low E for practicing, and then once you feel good with it, try some standard tuning chords. I play a few staccato beats not muted, then I mute with the left hand as sort of a punctuation to give definition to the beat.