I can understand your dilemma, and I also see from the Bible that unbelievers are not only apprehensive around born-again Christians due to some perceived fear of being judged, but their conscience will naturally convict them when any Christian is in their presence, since the unbeliever is following a different spirit, and a different Father.
It's in the nature of unbelievers to fight, balk, argue claw and bite, against all perceived Truth, this Truth is the 'light of God', that they are wrong and God is right. The Holy Ghost according to John 16:8 is dealing with all unbelievers about their sin. Before a sinner ever attends Church, The Spirit works through the life and 'light' of believers to bring conviction.
The unbeliever in most cases attends a Church service after he has come under conviction, this is due to the 'light' (Spirit of Truth) working inside of believers.
In Eph 2:2 we see that the unbeliever is following a spirit of disobedience, and in John 8:44, Jesus made it very plain that unbelievers are doing the lust of their father the devil.
All born-again Christians have been given and made 'light' by the Spirit of Truth who resides within them. John 1:4-9, Mat 5:14, 1 Thessalonians 5:5
Everyday when the 'light' (Christians) is in the presence of unbelievers, they will be dealt with by the Spirit of Truth who resides within us, they may never attend a Church service, but the Holy Ghost will preach to their conscience every time they are around a Christian. Especially if said Christian has been living a righteous life, for the unbeliever will read the pages of our behavior, that is contrary to their lifestyle.
Jesus said that these type people, who willingly live in sin and rebellion to 'light', refuse to come to the light because their deeds are evil. John 3:20
It takes 'conviction' from God, before the unbeliever stops fighting against God to come to his senses.
Be aware also, that there are many 'unseen' enemies of God who will constantly seek to attack, and bombard you with thoughts that you are at fault for being 'light', and therefore try to make you feel judgmental, for bringing 'conviction' to those God is trying to save.
I've found in many cases, it's not so much any perceived perceptions that may lead one to misapprehend the thoughts of unbelievers, but the constant mind attacks of the enemy.
(Wearing the Armor of God is a 24/7 obligation, as is to WATCH AND PRAY)
Personally I feel one of the best things a believer can do, when he thinks unbelievers are getting convicted around him/her, is SMILE..show them the joy of the Lord, and that your Father the True and Living God is better than their father the Devil.
The last thing a Christian should do is feel sorry that an unbeliever is under conviction! and worse yet, try and make them feel better. The 'feel better' comes when they 'repent' and give their hearts to God.
I've learned by experience, that one problem that hinders the 'light' from thoroughly bringing men to conviction, is when some Christians quench the Spirit that is dealing with the conscience of unbelievers, by trying to make them feel better when God is clearly dealing with them.
They will be convicted, look and feel sad, pitiful, angry, bitter, bad attitude, callous, around believers because God is at work, so whatever you do, don't quench the Spirit who is bringing the 'light' of Truth to their conscience.
I may be wrong, but I think some perceptions about being 'judgmental' may be motivated by an unwarranted need of being to sentimental about how unbelievers perceive us.
Sometimes the believer (I'm guilty of this) loses sight that they are ALWAYS in a spiritual war, this will surely be felt when we are around those that are still under the powers of darkness - just be careful one does not comprehend a feeling of being 'judgmental', as the Power of Christ working through them to destroy the powers of darkness.