I don't know much about Lordship Salvation, only enough to know that it appears to be a recognition of something often missing or wrongly understood or emphasized in parts of Protestant theology and an attempt to get back to historic Christianity-without necessarily realizing it I assume.
Yes, one telling sign that we have eternal life is by the way we act.
“Dear children, do not let anyone lead you astray. The one who does what is right is righteous, just as He is righteous. The one who does what is sinful is of the devil, because the devil has been sinning from the beginning. The reason the Son of God appeared was to destroy the devil’s work.” 1 John 3:7-8
Properly understood this is no different in kind than Jesus words to the young man in Matt 9:17-19
If you want to enter life, keep the commandments.”
“Which ones?” he inquired.
Jesus replied, “‘You shall not murder, you shall not commit adultery, you shall not steal, you shall not give false testimony, 1honor your father and mother,’ and ‘love your neighbor as yourself.’
All of this and more are summed up by-are accomplished by-the fulfillment of the greatest commandments, to love God with our whole heart, soul, mind, and strength and our neighbor as ourselves. That's to walk by the Spirit, who, alone, can cause or work that love in us.
"And hope does not put us to shame, because God’s love has been poured out into our hearts through the Holy Spirit, who has been given to us." Rom 5:5
That's the true definition of righteousness and in any case demonstrates that righteousness is a gift, that we must embrace and act on, not merely an imputation.
"...not having my own righteousness from the law, but that which is through faith in Christ, the righteousness from God on the basis of faith" (Phil 3:9).
"And so he condemned sin in the flesh, in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit." Rom 8:4
Again, as the church of Christ can so wisely teach:
"At the evening of life we shall be judged on our love."
And that proceeds from a life of walking in the Spirit.