Integrity while waiting involves obedience and good works…
Integrity defined: The entire, unimpaired state of any thing, particularly of the mind; moral soundness or purity; incorruptness; uprightness; honesty.
"Integrity is doing the right thing when you don’t have to—when no one else is looking or will ever know—when there will be no congratulations or recognition for having done so. Where is the man who will do the right thing, no matter what the cost?..." - Charles Marshall
Integrity is doing the right thing even when no one is there to see it.
Truth be told, God see’s it and He ought to be the reason for everything we do so everything we do should be done in accordance to His standards and according to his good pleasure.
In simpler terms, always choose what is right and reject what is wrong. This means avoiding sin and remaining above reproach for the glory of God.
In all things shewing thyself a pattern of good works: in doctrine shewing uncorruptness, gravity, sincerity,  Sound speech, that cannot be condemned; that he that is of the contrary part may be ashamed, having no evil thing to say of you. Titus 2:7-8
Wherefore, my beloved, as ye have always obeyed, not as in my presence only, but now much more in my absence, work out your own salvation with fear and trembling. For it is God which worketh in you both to will and to do of his good pleasure. Philippians 2:12-13
Wherefore gird up the loins of your mind, be sober, and hope to the end for the grace that is to be brought unto you at the revelation of Jesus Christ;  As obedient children, not fashioning yourselves according to the former lusts in your ignorance:  But as he which hath called you is holy, so be ye holy in all manner of conversation;  Because it is written, Be ye holy; for I am holy. 1 Peter 1:13-16
We should be diligent in being spotlessly and blamelessly ready for Jesus’ return at all times.
But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up. Seeing then that all these things shall be dissolved, what manner of persons ought ye to be in all holy conversation and godliness, Looking for and hasting unto the coming of the day of God, wherein the heavens being on fire shall be dissolved, and the elements shall melt with fervent heat? Nevertheless we, according to his promise, look for new heavens and a new earth, wherein dwelleth righteousness. Wherefore, beloved, seeing that ye look for such things, be diligent that ye may be found of him in peace, without spot, and blameless. 2 Peter 3:10-14
Keeping our lamps full and lit requires diligent effort on our part while we wait for our Bridegroom’s return.
Then shall the kingdom of heaven be likened unto ten virgins, which took their lamps, and went forth to meet the bridegroom. And five of them were wise, and five were foolish. They that were foolish took their lamps, and took no oil with them: But the wise took oil in their vessels with their lamps. While the bridegroom tarried, they all slumbered and slept. And at midnight there was a cry made, Behold, the bridegroom cometh; go ye out to meet him. Then all those virgins arose, and trimmed their lamps. And the foolish said unto the wise, Give us of your oil; for our lamps are gone out. But the wise answered, saying, Not so ; lest there be not enough for us and you: but go ye rather to them that sell, and buy for yourselves. And while they went to buy, the bridegroom came; and they that were ready went in with him to the marriage: and the door was shut. Afterward came also the other virgins, saying, Lord, Lord, open to us. But he answered and said, Verily I say unto you, I know you not. Watch therefore, for ye know neither the day nor the hour wherein the Son of man cometh. Matthew 25:1-13
Ye are the light of the world. A city that is set on an hill cannot be hid. Neither do men light a candle, and put it under a bushel, but on a candlestick; and it giveth light unto all that are in the house. Let your light so shine before men, that they may see your good works, and glorify your Father which is in heaven. Matthew 5:14-16
This reminds me, in a way, when we were in the process of foster/adopting our little ones. During this process we never knew when CPS would pop in for a visit, so we were extra diligent in keeping the house clean and in order. The potential for us to be able to adopt our little ones always hinged on us following the rules and being in a ready state for whenever CPS checked in on us. It is CPS that makes the final decision really on whether a family is fit to adopt. It is CPS’ assessment that the judge looks to when it comes to finalizing any adoption. In this manner it is the same with us waiting on our Lord and Saviour Jesus Christ to return. Thus, we remained in a ready state at all times and we did so without exception.
Being ready does not include sitting around and doing nothing especially when something good could and should be done because inaction in those cases is sin.
Godly integrity is keeping your lamps full so that God can receive glory. This requires action on our part that involves good works.
Therefore to him that knoweth to do good, and doeth it not, to him it is sin. James 4:17
When you do good to the “least of these” you are doing it as unto the LORD. When you do not do good to the “least of these” you go to hell. This ought to be motivation enough, especially if you know and fear the LORD.
Then shall the King say unto them on his right hand, Come, ye blessed of my Father, inherit the kingdom prepared for you from the foundation of the world: For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in: Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me. Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee ? or thirsty, and gave thee drink? When saw we thee a stranger, and took thee in? or naked, and clothed thee ? Or when saw we thee sick, or in prison, and came unto thee? And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me. Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels: For I was an hungred, and ye gave me no meat: I was thirsty, and ye gave me no drink: I was a stranger, and ye took me not in: naked, and ye clothed me not: sick, and in prison, and ye visited me not. Then shall they also answer him, saying, Lord, when saw we thee an hungred, or athirst, or a stranger, or naked, or sick, or in prison, and did not minister unto thee? Then shall he answer them, saying, Verily I say unto you, Inasmuch as ye did it not to one of the least of these, ye did it not to me. And these shall go away into everlasting punishment: but the righteous into life eternal. Matthew 25:34-46
Pure religion and undefiled before God and the Father is this, To visit the fatherless and widows in their affliction, and to keep himself unspotted from the world. James 1:27
And whatsoever ye do in word or deed, do all in the name of the Lord Jesus, giving thanks to God and the Father by him. Colossians 3:17
Integrity while waiting involves obedience and good works.
A true believer’s perspective on this matter is not that we ‘have to’ but that we ‘get to’ do good for His glory!
So, go on, do good even when it is just God watching. He loves it when we obey Him and do good all for His glory.