[MC5C]

An awful lot of jazz players use a pick, but hybrid with pick and fingers for comping or "popping" chords. I use pick and three fingers for chords. Rarely do jazz players strum the chords in chord melody, but strumming with a pick is very common in rhythm comping work. Don't worry so much about damping the strings you don't need, just finger the chord and let nature take it's course. If you are rhythm comping, just play a few notes of the chord, don't worry about getting every note in. Freddie Green played a lot of one (!) and two note chords. If you are chord melody, try focusing on the melody note and just add one or two other notes that sound nice. They don't have to be a formal "that's a chord out of a book" chord, they just have to sound nice and imply the harmony. A melody note, a root and a third almost always sounds nice. You have your harmonic base, your major or minor, and your melody note. I play one tune where the melody is a flat 9 of the root, so I can play the root, a minor third and the flat nine, and I have the key elements of the m7b9 chord that is the actual chord for that beat.

You can leave out a lot of the notes in a chord, including the root if the root is implied by the harmony. For example, in a II/V/I, you can leave out the root of the V chord, which is a dominant chord, and play something like a 7/9/13 chord thusly: for G 7/9/13, play F on 4th string, B on 3rd string, E on 2nd string and A on 1st string. You have the 7th, 3rd, 13th and 9th. It's a great chord, very easy to play. If you are playing with a pick and don't want to use your fingers as well, try playing four note chords just using the DGBE strings and don't use the EA strings at all. Or the same thing just using the middle four strings and never playing the E strings. Very interesting to try, opens up new ideas.