TITUS 3:5 not by works of righteousness which we have done, but according to His mercy He saved us, through the washing of regeneration and renewing of the Holy Spirit.
This verse needs to be interpreted not just by analysing the verse, but also by using the wider context of all the New Testament epistles.
I (personally) am shocked by the NLT translation, it clearly seems to have been interpreted from a doctrinal foundation. The word ‘sin’ does not appear, neither is it implied!
The Washing (λουτρόν, loutron) implies fully bathed, as in baptism. And, believers are all ‘baptised’ into Christ. Then looking at, and including Regeneration (παλιγγενεσία palingenesía), this clearly is a reference to the new birth - Strongs has its meaning as ‘new birth, reproduction, renewal, recreation, regeneration’.
So, ‘washing of regeneration’ simply means that through being born again, we are ‘baptised’ into Christ. This ‘process’ is also clearly outlined in Paul’s letters, that is, it ‘fits’ with the rest of what is taught in the New Testament.
The context, taken from the start of this verse is righteousness - which can not come from ‘our’ works, neither the Law, but comes through our new birth, through ‘regeneration’.
Now let’s consider ‘renewing of the Holy Spirit.’ The New Testament clearly outlines the ministry of the Holy Spirit that is, what ‘He’ does..
JOHN 14:26 But the Helper, the Holy Spirit, whom the Father will send in My name, He will teach you all things [snip]
Now, with this in mind, consider the following...
ROMANS 12:2 And do not be conformed to this world, but be transformed by the renewing of your mind, [snip]
So now we can see, from scripture, what renewing of the Holy Spirit.’ refers to.
Now to the Q - ‘what is the ‘and’ joining’, the two parts of this verse in Titus ... This verse is about being saved. We are ‘saved’ by being ‘born again, regenerated. Our soul is saved by the renewal through the Holy Spirit. (He restores my soul - psalm 23). One is ‘instant’, and results in eternal life. The other is a life long process, for living in this life.
So in summary, we are saved through regeneration (new birth), and then (after that), we (our ‘soul/mind/thinking’) are ‘renewed’ through the work of the Holy Spirit.