First off, occasional misses in alternate bass strumming are kind of normal, it happens to the best of them
Slowing down is good, but only as far as makes sense. You should practice slightly below your limit to gradually extend your "comfort zone".
What helped me with this when I started out was doing picking excercises to learn to hit specific strings, e.g. (string numbers) 6 5 6 4 6 3 6 2 6 1 and back again, or 6 1 5 1 4 1 3 1 2 1 and back. You can make them up as you go along (variety is key, you're practicing accuracy, not learning a pattern).
The rest of it is really just practice. You need to get your picking hand to pick the correct root note without thinking much about it.
One useful trick: like in normal strumming, try muting the 6th string with your thumb when it's not wanted, esp. when it's a chord that doesn't contain the note E. That way, if e.g. you're playing a D chord an accidentally hitting the low E, you get a little "thud" at most, but you don't turn the chord into something different.