Part of our admin training at work this year was the idea of “thank, don’t apologize”. Say “thank you for helping with [x]”, not “sorry you had to do [x]”. Thanking people makes them feel appreciated, while apologizing makes something (either your actions or the job) look bad, which doesn’t encourage positive feelings.
Also, you’d be shocked how far you can get with coworkers by modeling good behavior. If your workplace is a hive of blame and finger-pointing, be the person that says “Thanks for the review – This one is on me. I’ll get the corrections out within [appropriate timeline].”
If you are going to make accusations, your evidence had better be airtight and you WILL be criticized, especially by the people you just threw under the bus instead of just admitting you screwed up. Your only allies are like-minded souls who will also throw you under all 18 wheels if it’s expedient.
If you are the person who is just plain Handling It, without drama, the people who matter are likely to be grateful and the people who were sharpening up their pointy finger have just had the wind kicked out of their bullshit. The friend you make by being the person who gets things taken care of might be the one writing a performance review or signing your paycheck, because those are the people who catch it when things don’t happen.