Yeah. But you have to keep in mind that Jazz doesn't use one particular scale throughout the entire composition. I'm sure you've heard the idea that Jazz music is composed of playing different scales over different chords as they progress. So you may use the Mixolydian scale over one chord then the next chord you may use the Dorian scale.

The melodic minor (jazz melodic minor) is just the minor scale with a raised 6th and 7th. To play jazz, a good head start is to take a common chord progressions, I-V-I-IV-I but use more advanced voicing. Two chords that sound jazzy per say are 6th chords and 13th chords.

When you say they have the same root, I'm guessing your using one scale over half the length of one chord, and another scale over the second half of that chord. Otherwise, you're going to have to change the root of the scale to match the root of the chord your playing over. This is what I meant about using different scales over different chords in jazz. When your looking for a compatible scale for a chord, all you have to look for is that the scale and the chord contain the same notes. So if the chord is a C major 7 chord, any scale with C, E, G, and B in it will work. As an example 2 scales that work with the same root would be C major and C Lydian. SO you could use both these scales over one C major 7th chord, or you could chose between either to be played over a C major 7th chord. Once the progression changes and the C major 7th goes to another chord, you will change the scale to fit the next chord (jazz).

If your looking for new scales to mess with, try checking out some of the more exotic scales. The Hungarian scales, gypsy scales, and eastern scale are intriguing.