Did Sauron end his service to Morgoth after he lost Tol Sirion or did he earn back his master's trust?

From reading the Silmarillion I had thought that Sauron disappeared and ended his service to Morgoth once he lost Tol Sirion to Luthien and Huan. However, after reading through the Lays of Leithian and thinking more about it I've come to the opposite conclusion and I would like to know what is the correct interpretation of events.

From the Lays of Leithian

With gasping breath and shuddering

he spake, and yielded as he must,

and vanquished betrayed his master's trust.

The above quote refers to Sauron yielding his power over the Wizard's Isle to Luthien, and being strangled by the impossible grip of Huan. It says here that he betrayed Morgoth's trust, and I must admit that it's difficult to conceive of Morgoth forgiving anybody of anything, unless of course he feigned forgiveness for evil purposes.

After this he flees and builds a fortress in Taur-na-Fuin. Again from the Lays of Leithian

A vampire shape with pinions vast

screeching leaped from the ground, and passed,

its dark blood dripping on the trees;

and Huan neath him lifeless sees

a wolvish corpse - for Thu had flown

to Taur-na-Fuin, a new throne

and darker stronghold there to build.

It says that his immediate plan was to build a stronghold that was even darker than the one he had twisted out of Tol Sirion. This sounds like he wanted to build the fortress for Morgoth; a fortress that was even darker and more terrible than before, one that Morgoth would appreciate so much that he would forgive his servant of his blunder of losing the Nolderin Tower at Tol Sirion.

I feel that Sauron being found at the end of the War of Wrath further lends credence to this interpretation. Why was Sauron found by Eonwe if he was not fighting Morgoth's war? If he had simply vanished and could not be found by Morgoth then what hope would Eonwe have of finding him? This seems to indicate that Sauron did not end his service to Morgoth after he lost Tol Sirion, and in fact was actively building a fortress in dedication to Morgoth.

Did he or did he not end his service to Morgoth after losing Tol Sirion?