Spritual integrity' is firmness of character -- integral with essential things; namely Christ (Col 2:3; Col 2:9; Col 2:10; Col 2:19); David exemplifies needed integrity (Psa 7:8-10; Psa 25:1-22; Psa 26:1-12; Psa 41:1-12 ). 5 things are demanded: Col 2:11we should live separate from the world (because of spiritual circumcision); Col 2:12 we should live in fellowship (because we're baptised into union with Him); Col 2:13 we should walk in newness of life (because we're raised with Him); Col 2:10 we should refuse to look elsewhere for fulfilment (because we're complete) ; Col 2:17 we should remain attached to the Head (because we are now in His Body).
Paul never met the Colossians but wrote to them from a Roman prison around 62 AD to counter misconceptions of his day. It magnifies the character of Jesus Christ more than any other book in the New Testament, and counters misconceptions of our day as well. The key verse may well be Col 1:18 "...that in all things, He might have preeminence."