Any advice for avoiding careless errors?
Yes! Slow down.
I say slow down, when what I really mean is relax. The difference, measured in seconds, between the amount of time it takes you to read a sentence:
"As to the issue of Justin Bieber's romantic inclinations, I could care less"
or to correctly read the sentence:
"As to the issue of Justin Bieber's romantic inclinations, I couldn't care less"
almost can't be measured.
Probably you can't measure it using your wristwatch, anyway. But that miniscule difference in time means the difference between an author caring about the Biebs' love life in the first instance, and not caring about the Biebs in the second.
To catch that distance, you just have to read, which shouldn't be hard, but sometimes is hard nonetheless. So my best advice to you is to try to slow down, just a bit. You have the time to read properly. Make yourself know that - where it counts - and your actions will start to reflect it.
I think that will help. Hope it does.