Would someone who didn't truly repent of their sins have the desire to serve the Lord?
For instance, would they even care that when they sin it offends the Lord?
And what are some of the signs of a false convert? Thanks!
People "serve the Lord" for reasons often quite divorced from a love for him. This was true of the Pharisees of Christ's time and it's true of "Christians" today. Religious pride, self-righteousness, the momentum of upbringing - these and many other things can all be (bad) inducements to "serve the Lord."
The Pharisees appeared to have no compunction about obeying the law of Moses in a deeply self-serving, hypocritical way, twisting that law through the additions of many man-made "traditions." I've met a great many "Christians" who appear to have no sensitivity whatever to the offense their carnality is to God. They are contentious, petty, quick to offense, constantly needing to be acknowledged in their "work for the Lord," bitter and short-tempered. In secret, many of them have issues with a variety of fleshly sins: addiction to porn, to food, to alcohol, to gambling, etc. Consequently, they have become skilled deceivers, liars, having first deceived themselves into sin and then deceiving others about their sin.
Anyway, what are signs of a false convert? See above. Also:
1.) They are occupied with fleshly gratification, loving, even, what should shame them, focused upon the here-and-now rather than eternity.
Philippians 3:18-19
18 For many, of whom I have often told you and now tell you even with tears, walk as enemies of the cross of Christ.
19 Their end is destruction, their god is their belly, and they glory in their shame, with minds set on earthly things.
2.) They depart the faith.
1 John 2:19
19 They went out from us, but they were not of us; for if they had been of us, they would have continued with us. But they went out, that it might become plain that they all are not of us.
3.) They hold to unbiblical doctrines, rebel against authority, and seek positions of advantage in the Church which they occupy in arrogance.
Jude 1:4, 8, 16
4 For certain people have crept in unnoticed who long ago were designated for this condemnation, ungodly people, who pervert the grace of our God into sensuality and deny our only Master and Lord, Jesus Christ.
8 Yet in like manner these people also, relying on their dreams, defile the flesh, reject authority, and blaspheme the glorious ones.
16 These are grumblers, finding fault, following after their own lusts; they speak arrogantly, flattering people for the sake of gaining an advantage.
You might also take a look at the following link which details the most common indicators that the Spirit genuinely indwells a person:
Am I Really Born Again?