The only reason some people "try" to make it look like the Holy Spirit was in Saul in the OT like He is in the believer in the New Covenant is to try to show the false notion that the Holy Spirit will leave us like Saul. Even though Jesus said the complete opposite.
He also left them.
Prophets and men of God were part of the Kingdom of God.
These men had a special relationship with God, but little is said how the Holy Spirit
communed with them.
Their covenant was sealed by circumcision of the flesh and the heart.
All were given equal opportunity, but some openly rebelled and died.
Until the full realisation of the church or body of Christ, things were not complete.
It was a mystery yet to come, a people with the law on their hearts.
But the shadow was still powerful and significant but not the final revelation of power.
It is simply not true that people are seeking to say the old and new covenant are the
same, because clearly one is different to the other. But they are not one is a failure
while the other a success, because one is the path towards the other, an indication and
structure of what it actually means.
So if we can agree on this we have progress, though I doubt this is possible.
And look at the church today. We are very much still human and flawed, and suffering
similar issues the Israelities fell into. Which probably also shows us the ways God expects
us to respond when these situations occur.