ASD72,
We are definitely children of God who, as you noted call out from the depths of our spirit, “Abba, Father”. So to answer your question, my hunch is this may be referring back to Exodus chapter 21: 1-7. A Hebrew slave, after serving a Hebrew master for 7 years, had the option to go free. However, after the 7th year, if the slave chose to stay out of love and fidelity to a kind and loving master who treated the slave as one of his own family members, he could do so. Also, in Psalm 19:12, and in various verses throughout Ps 119 the psalmist refers to himself as a servant (vs 17, 23,38,and 49 to name a few). These verses show the humility and trusting relationship the psalmist had with the Lord, acknowledging God’s great love and mercy while being fully mindful of the infinite majesty of our Lord…
In the New Testament, we find Jesus Himself taking on the form of a slave (see Philippians 2:5-11) in that He becomes our brother in taking on the the burden of our sin and joins us in full solidarity as being not only God, but fully human, experiencing the pain and suffering sin brings. We are in a sense, slaves to sin in that we experience all of the pain and suffering of sin, and are still battling its affects in our being (See Romans chapter 6)…
So, when the Apostles refer to themselves as slaves of the Lord, it is not the type of relationship that exists in cruel, cold, chattel slavery, it is really a beautiful picture of one who is stating, if you will, “You have given your all for me, yet is really I who owe my all for you, therefore, as you gave yourself as a sacrifice for me, I submit myself totally to you as a servant, yet knowing fully you see me as your child.” (Or something like that…

) Hope this helps a bit.