First off, you might need to raise your action for slide work. If it's buzzing at the point of the slide you might not be pressing hard enough, but even typical lower action reduces the room for error in this aspect, causing other fret buzzes. These changes might also require an intonation set-up too.

The other important thing is hand muting, by both hands. With the slide hand, lay the other unused fingers behind the slide resting on the strings. Same thing with picking hand...unused fingers/palm muting the strings. You might also try fingerpicking, to free up even more of your fingers/hand for muting purposes.

You might also browse through these lessons (if you haven't already) for other tips and tricks.