that's because you use random downstroke/upstroke combination I guess. practice makes perfect. the best way is to relax yourself, and start by singing to yourself at first. make sure you sing and strum according to the tempo.

after that you can concentrate on pitching and stuffs

p/s: can't imagine Dave Mustaine singing while riffing the song "Holy Wars" 

Like soulfly has pointed out, practice makes perfect.
Try practicing the singing & strumming seperately and master them, in the sense that you know the tune inside out (When to come in during the verse, what patterns to strum etc...). Then try out both simultaneously with a slower tempo. When you start to get the hang of it, push the tempo up to the studio speed & you should be able to fare better this time. Again... practice is the key.

I'm not good at singing and playing yet, still got a long way to master it.
i can give some tips(stuff i do)

what i do is i put the name of the chord on top of the lyrics so i know when to change chord according to the song(eventually you'll 'feel' it)
if you practice that way, you will eventually know when to change the chord, like on which word of the lyric.

another thing is to practice each part seperately. for me i find playing while distracted is a good practice such as watching tv while you strum sub consciously or ask your friend to give you simple math sums 

hope this helps.