I believe that O.S.A.S. is a misnomer, and with the Puritans believe rather in the Perseverance of the Saints. This is from the Baptist 1689 London Confession, XVII: 2-3:
This perseverance of the saints does not depend on them- that is, on their own free will. It rests upon the immutability of the decree of election [Romans 8:30; 9:11-16], which flows from the free and unchangeable love of God the Father. It also rests upon the efficacy of the merit and intercession of Jesus Christ and upon the union which true saints have with Him [Romans 5:9-10; John 14:19].
- It rests upon the oath of God [Hebrews 6:17-18], and upon the abiding of His Spirit.
- It depends upon the seed of God being within them [1 John 3:9] and upon the very nature of the covenant of grace [Jeremiah 32:40].
-All these factors give rise to the certainty and infallibility of the security and perseverance of the saints.
The saints may, through the temptation of Satan and the world, and because their remaining sinful tendencies prevail over them, and through their neglect of the means which God has provided to keep them fall into grievous sins.
They may continue in this state for some time [Matthew 26:70-74] , so that they incur God's displeasure, grieve His Holy Spirit [Isaiah 64:5-9; Ephesians 4:30], suffer the impairment of their graces and comforts [Psalm 51:10-12], have their hearts hardened and their consciences wounded [Psalm 2:3-4], and hurt and scandalise others. By this they will bring temporal judgements upon themselves [2 Samuel 12:14].
Yet they shall [in time] renew their repentance and be preserved, through faith in Christ Jesus, to the end [Luke 22:32, 61-62].