No one says “don’t judge me” when they are proud of something they’ve done. It is never used to express modesty. You don’t hear anyone say “don’t judge me for giving all of my money to the poor”.
The phrase is used exclusively when someone feels guilty about having done something that is obviously wrong. It is almost always used in a situation where sound judgment, spoken in good faith, would be the best possible thing to ask for.
Christians of all people should know better than the use the phrase, but as in many things, many of the people who claim to be Christ’s followers resemble the world around them more than their Lord. The Bible does not unilaterally condemn judgment. The oft cited verses (Matthew 7:1 and Luke 6:37) that say “do not judge, or you too will be judged” are never cited in full context. Christ qualifies the statement by telling His disciples that they ought to make sure that they apply the same standard to themselves that they apply to others. He says at another time (John 7:24) to “judge with righteous judgment”.
Without the ability to judge the actions of other people, we wouldn’t be able to have a legal system. It would be impossible to enforce laws. You wouldn’t be able to avoid a known serial killer because to even call him a serial killer is an act of judgment. You wouldn’t be able to lock your doors at night lest you judge the sort of people who might try to enter your home.
Whenever you are tempted to say “don’t judge me”, it would be wiser to instead stop doing the act you don’t want judged. Chances are, if it’s worth doing, you wouldn’t be asking for people to overlook it.