Sauron's loyalty after the First Age

[Zigûr]

Originally Posted by Mumriken

This loyalty exists in all beings that are dependent on another being. It's actually a form of love the orcs have for Sauron.

The Orcs served Sauron out of fear, not love. They fear him (and the Nazgûl) more than they feared their enemies.

Even if Morgoth is not there to give him power he still honors Morgoth by trying to follow in his footsteps even though he is unable to do so because of not being as powerful as him.

If it was written somewhere that Sauron thought he was doing this deliberately to honour Morgoth I would agree with you. I guess I just don't believe that you can be "accidentally loyal". Loyalty in my opinion involves some element of deliberate decision-making.

Probably the reason he did not fully destroy the children as Morgoth tried to do was because he simply wasn't powerful enough to do so.

I'm afraid Professor Tolkien covered Sauron's lack of destructive intent:

the only real good in, or rational motive for, all this ordering and planning and organization was the good of all the inhabitants of Arda (even admitting Sauron’s right to be their supreme lord)

Also:

Sauron had never reached this stage of nihilistic madness. He did not object to the existence of the world, so long as he could do what he liked with it. ~Morgoth's Ring.

Sauron wanted to rule Middle-earth and its inhabitants. Whether or not he had the power to destroy it/them isn't especially relevant: even if he could, he didn't want to.